WEBVTT 00:00:00.000 --> 00:00:07.760 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:07.760 --> 00:00:09.897 align:middle line:90% 00:00:09.897 --> 00:00:11.480 align:middle line:84% DAVID KAISER: So today, we are talking 00:00:11.480 --> 00:00:14.330 align:middle line:84% about the reception of Einstein's work 00:00:14.330 --> 00:00:17.575 align:middle line:84% from 1905 on the electrodynamics of moving bodies. 00:00:17.575 --> 00:00:18.950 align:middle line:84% So we'll talk about the reception 00:00:18.950 --> 00:00:23.880 align:middle line:84% of what we have come to call the Special Theory of Relativity. 00:00:23.880 --> 00:00:27.170 align:middle line:84% So as a quick reminder, we're going to-- 00:00:27.170 --> 00:00:28.040 align:middle line:90% sorry, excuse me. 00:00:28.040 --> 00:00:31.130 align:middle line:84% The class today has three main parts, like they usually do. 00:00:31.130 --> 00:00:33.110 align:middle line:84% We'll talk about some of the early reactions 00:00:33.110 --> 00:00:36.090 align:middle line:84% to Einstein himself and to this body of work. 00:00:36.090 --> 00:00:38.410 align:middle line:84% And then we'll look at two specific examples 00:00:38.410 --> 00:00:41.060 align:middle line:84% of those reactions in a bit more detail. 00:00:41.060 --> 00:00:43.250 align:middle line:84% Get more of a sense of what ways were 00:00:43.250 --> 00:00:46.850 align:middle line:84% people could have engaging with Einstein's paper from 1905, 00:00:46.850 --> 00:00:48.920 align:middle line:84% what did they think it was really about, 00:00:48.920 --> 00:00:50.480 align:middle line:84% what did they get excited about, what 00:00:50.480 --> 00:00:52.820 align:middle line:84% did they ignore that otherwise seem to have been really 00:00:52.820 --> 00:00:54.210 align:middle line:90% important to Einstein himself. 00:00:54.210 --> 00:00:57.440 align:middle line:84% So we're going to talk about not just did people read the paper 00:00:57.440 --> 00:01:00.150 align:middle line:84% and make use of it, but in what specific ways? 00:01:00.150 --> 00:01:03.350 align:middle line:84% How were they creatively making this work 00:01:03.350 --> 00:01:06.440 align:middle line:84% relevant from where they sat intellectually and 00:01:06.440 --> 00:01:07.430 align:middle line:90% institutionally? 00:01:07.430 --> 00:01:09.930 align:middle line:84% And we'll see that sometimes that that diverged-- 00:01:09.930 --> 00:01:11.783 align:middle line:90% sometimes in dramatic ways-- 00:01:11.783 --> 00:01:13.200 align:middle line:84% from what Einstein himself thought 00:01:13.200 --> 00:01:15.930 align:middle line:84% was most relevant or most interesting or most important 00:01:15.930 --> 00:01:16.740 align:middle line:90% about his own work. 00:01:16.740 --> 00:01:19.080 align:middle line:84% And that's a theme again we've seen already a few times 00:01:19.080 --> 00:01:20.717 align:middle line:90% so far in this class. 00:01:20.717 --> 00:01:22.800 align:middle line:84% Different people engaging with Maxwell's equations 00:01:22.800 --> 00:01:24.173 align:middle line:90% in different ways. 00:01:24.173 --> 00:01:25.590 align:middle line:84% It's a theme that we'll see really 00:01:25.590 --> 00:01:27.340 align:middle line:90% throughout the entire semester. 00:01:27.340 --> 00:01:28.728 align:middle line:90% And here's a class I like-- 00:01:28.728 --> 00:01:30.270 align:middle line:84% this is pretty fun material because I 00:01:30.270 --> 00:01:33.930 align:middle line:84% think we get to really sit with some examples in some detail 00:01:33.930 --> 00:01:35.790 align:middle line:84% and really watch people wrestle with what 00:01:35.790 --> 00:01:39.830 align:middle line:84% was a fairly new material, Einstein's paper from 1905. 00:01:39.830 --> 00:01:40.910 align:middle line:90% So just a reminder. 00:01:40.910 --> 00:01:43.160 align:middle line:84% What did we cover in our previous discussion 00:01:43.160 --> 00:01:44.810 align:middle line:90% just on Monday? 00:01:44.810 --> 00:01:47.450 align:middle line:84% Einstein had many different threads 00:01:47.450 --> 00:01:50.930 align:middle line:84% that he brought together in this paper from 1905 entitled, 00:01:50.930 --> 00:01:53.390 align:middle line:84% "On the Electrodynamics of Moving Bodies." 00:01:53.390 --> 00:01:56.280 align:middle line:84% He was a recent college graduate. 00:01:56.280 --> 00:01:59.960 align:middle line:84% He was working off-hours on his PhD thesis. 00:01:59.960 --> 00:02:02.360 align:middle line:84% He had a full-time day job working 00:02:02.360 --> 00:02:05.180 align:middle line:84% at the patent office in Bern, Switzerland 00:02:05.180 --> 00:02:08.660 align:middle line:84% at the lowest level, entry-level job of patent officer 00:02:08.660 --> 00:02:09.860 align:middle line:90% third class. 00:02:09.860 --> 00:02:12.650 align:middle line:84% And in the evenings, he would hang out with his friends 00:02:12.650 --> 00:02:16.070 align:middle line:84% like Maurice Sullivan and Conrad Habicht 00:02:16.070 --> 00:02:20.180 align:middle line:84% and another friend, Michelangelo Besso, in a small circle. 00:02:20.180 --> 00:02:22.950 align:middle line:84% And they would read physics and philosophy together, 00:02:22.950 --> 00:02:24.950 align:middle line:84% and they called themselves the Olympia Academy 00:02:24.950 --> 00:02:27.533 align:middle line:84% even though it was really just three recent graduates drinking 00:02:27.533 --> 00:02:30.050 align:middle line:84% beer in the pub talking about things 00:02:30.050 --> 00:02:31.840 align:middle line:90% like the work of Ernst Mach. 00:02:31.840 --> 00:02:34.970 align:middle line:84% It's not a bad way to spend your time after college. 00:02:34.970 --> 00:02:37.850 align:middle line:84% So one of the people they were really interested in was Ernst 00:02:37.850 --> 00:02:40.830 align:middle line:84% Mach, and again, we have this great reading that was 00:02:40.830 --> 00:02:43.770 align:middle line:84% on the Canvas site I think designed for this 00:02:43.770 --> 00:02:46.710 align:middle line:84% for Monday's-- this past Monday's class by our own 00:02:46.710 --> 00:02:49.920 align:middle line:84% Amanda Gefter, one of our TAs, who helps us really even more 00:02:49.920 --> 00:02:51.810 align:middle line:84% understand what was Mach all about, 00:02:51.810 --> 00:02:54.160 align:middle line:84% what did Einstein take from Mach's work? 00:02:54.160 --> 00:02:56.730 align:middle line:84% And so I encourage you to go back and reread Amanda's piece 00:02:56.730 --> 00:02:57.640 align:middle line:90% as well. 00:02:57.640 --> 00:03:00.720 align:middle line:84% So Ernst Mach is this Viennese polymath 00:03:00.720 --> 00:03:02.850 align:middle line:84% who introduced this philosophy of science 00:03:02.850 --> 00:03:05.730 align:middle line:84% in the later years of the 19th century, 00:03:05.730 --> 00:03:08.790 align:middle line:84% really during Einstein's childhood, that 00:03:08.790 --> 00:03:10.470 align:middle line:90% came to be called positivism. 00:03:10.470 --> 00:03:13.050 align:middle line:84% And as far as Ernst Mach was concerned, 00:03:13.050 --> 00:03:15.540 align:middle line:84% the best way to make progress scientifically 00:03:15.540 --> 00:03:18.445 align:middle line:84% was to focus on objects of positive experience-- 00:03:18.445 --> 00:03:20.070 align:middle line:84% that's where the positivism comes from. 00:03:20.070 --> 00:03:23.790 align:middle line:84% Things we could really concretely observe or measure. 00:03:23.790 --> 00:03:27.370 align:middle line:84% And not focus on things that could never, even in principle, 00:03:27.370 --> 00:03:30.180 align:middle line:84% be subject to these kind of empirical inputs. 00:03:30.180 --> 00:03:36.780 align:middle line:84% So we had this extensive, often very biting, very significant 00:03:36.780 --> 00:03:39.270 align:middle line:84% critique of Newtonian physics, which 00:03:39.270 --> 00:03:41.220 align:middle line:84% Mach was-- as far as Mach was concerned, 00:03:41.220 --> 00:03:44.010 align:middle line:84% was riddled with these things that could never ever become 00:03:44.010 --> 00:03:46.210 align:middle line:90% objects of positive experience. 00:03:46.210 --> 00:03:48.570 align:middle line:84% And that really was very, very inspiring 00:03:48.570 --> 00:03:50.790 align:middle line:84% for young Einstein and his circle of friends. 00:03:50.790 --> 00:03:53.250 align:middle line:84% So one of the lessons that Einstein took from that, 00:03:53.250 --> 00:03:57.360 align:middle line:84% among several, in these years leading up to around 1905 00:03:57.360 --> 00:04:00.060 align:middle line:84% was that it would make sense to focus on kinematics, 00:04:00.060 --> 00:04:02.703 align:middle line:84% the motion of objects through space and time, 00:04:02.703 --> 00:04:04.620 align:middle line:84% things we could, at least in principle, really 00:04:04.620 --> 00:04:07.290 align:middle line:84% measured and observed, rather than starting 00:04:07.290 --> 00:04:09.810 align:middle line:84% our inquiry with dynamics, the study of forces, that 00:04:09.810 --> 00:04:12.510 align:middle line:84% was really turning on its head, what had become quite 00:04:12.510 --> 00:04:15.720 align:middle line:84% a standard way to approach topics like electrodynamics. 00:04:15.720 --> 00:04:18.390 align:middle line:84% We saw when Hendrik Lorentz considered electrodynamics. 00:04:18.390 --> 00:04:21.300 align:middle line:84% He would start with the forces exerted by the ether on objects 00:04:21.300 --> 00:04:21.930 align:middle line:90% and so on. 00:04:21.930 --> 00:04:24.750 align:middle line:84% Einstein said, let's start with kinematics first. 00:04:24.750 --> 00:04:27.600 align:middle line:84% It was that kind of thinking that led him to postulate 00:04:27.600 --> 00:04:32.040 align:middle line:84% these opening postulates in his 1905 paper, which hopefully 00:04:32.040 --> 00:04:34.570 align:middle line:84% is now quite familiar to you in these opening few pages. 00:04:34.570 --> 00:04:37.620 align:middle line:84% This is the subject for your first paper assignment. 00:04:37.620 --> 00:04:40.290 align:middle line:84% And to paraphrase these two postulates 00:04:40.290 --> 00:04:43.270 align:middle line:84% that Einstein elevates in those opening paragraphs, 00:04:43.270 --> 00:04:47.010 align:middle line:84% the first is a extension or generalization 00:04:47.010 --> 00:04:49.860 align:middle line:84% of an idea that was actually centuries old by this point. 00:04:49.860 --> 00:04:51.930 align:middle line:84% He was really building on Galileo's work. 00:04:51.930 --> 00:04:55.050 align:middle line:84% And Einstein says, not only should the laws of mechanics 00:04:55.050 --> 00:04:58.380 align:middle line:84% be valid for any observer who is moving at a constant speed, 00:04:58.380 --> 00:05:00.600 align:middle line:84% as long as they're not speeding up or slowing down-- 00:05:00.600 --> 00:05:02.880 align:middle line:84% they're not accelerating, then all the laws 00:05:02.880 --> 00:05:06.120 align:middle line:84% of mechanics of ballistics of a ball 00:05:06.120 --> 00:05:08.438 align:middle line:84% you might toss and play catch, all those laws 00:05:08.438 --> 00:05:10.980 align:middle line:84% are valid whether you're on a boat moving at a constant speed 00:05:10.980 --> 00:05:13.873 align:middle line:84% down the river or standing still on the shore. 00:05:13.873 --> 00:05:15.540 align:middle line:84% Einstein says, that's true-- that should 00:05:15.540 --> 00:05:16.860 align:middle line:90% be true for all of physics. 00:05:16.860 --> 00:05:19.990 align:middle line:84% For electricity, for magnetism, for optics, for thermodynamics. 00:05:19.990 --> 00:05:22.860 align:middle line:84% So part 1 is elevating or generalizing 00:05:22.860 --> 00:05:25.350 align:middle line:84% what was already a fairly familiar notion 00:05:25.350 --> 00:05:27.000 align:middle line:90% to many, many physicists. 00:05:27.000 --> 00:05:29.250 align:middle line:84% And part two, at least at first, he says, 00:05:29.250 --> 00:05:32.842 align:middle line:84% it looks apparently irreconcilable with the first. 00:05:32.842 --> 00:05:35.400 align:middle line:84% The second postulate, according to Einstein, 00:05:35.400 --> 00:05:39.030 align:middle line:84% is that the speed of light, c, is a constant independent 00:05:39.030 --> 00:05:40.620 align:middle line:90% of the motion of the source. 00:05:40.620 --> 00:05:42.360 align:middle line:90% And we now know-- 00:05:42.360 --> 00:05:45.750 align:middle line:84% he came to that in his own private thinking, 00:05:45.750 --> 00:05:48.120 align:middle line:84% or in talking with a small circle of friends-- 00:05:48.120 --> 00:05:50.048 align:middle line:84% by a series of these thought experiments like, 00:05:50.048 --> 00:05:51.840 align:middle line:84% what would it be like if you could catch up 00:05:51.840 --> 00:05:54.120 align:middle line:84% to a light wave like this surfer riding 00:05:54.120 --> 00:05:56.130 align:middle line:84% at the same speed as the ocean wave? 00:05:56.130 --> 00:05:58.890 align:middle line:84% You would see this frozen field configuration 00:05:58.890 --> 00:06:01.600 align:middle line:84% which, Einstein was convinced, could never really happen. 00:06:01.600 --> 00:06:04.290 align:middle line:84% So how do he make sure no one gets caught in that absurdity, 00:06:04.290 --> 00:06:07.080 align:middle line:84% makes sure no one could ever catch up to a light wave? 00:06:07.080 --> 00:06:09.480 align:middle line:84% Well, if light's always going to travel at a fixed speed 00:06:09.480 --> 00:06:12.090 align:middle line:84% no matter how fast you're chasing after it. 00:06:12.090 --> 00:06:15.420 align:middle line:84% If I'm on a supersonic jet plane, 00:06:15.420 --> 00:06:17.820 align:middle line:84% I'm still going to see the light wave travel past me 00:06:17.820 --> 00:06:19.380 align:middle line:84% at this fixed speed of light, then 00:06:19.380 --> 00:06:22.650 align:middle line:84% no matter what fancy engines I sup up 00:06:22.650 --> 00:06:26.100 align:middle line:84% on my means of transportation, I will never, ever 00:06:26.100 --> 00:06:27.930 align:middle line:84% be like this surfer, I will never 00:06:27.930 --> 00:06:31.110 align:middle line:84% catch up to and move at the same speed as that light wave. 00:06:31.110 --> 00:06:34.020 align:middle line:84% And that was, Einstein considered, really central 00:06:34.020 --> 00:06:35.040 align:middle line:90% as a postulate. 00:06:35.040 --> 00:06:37.020 align:middle line:84% He doesn't prove it, he doesn't even really 00:06:37.020 --> 00:06:39.410 align:middle line:84% motivate it in this paper very strongly. 00:06:39.410 --> 00:06:43.420 align:middle line:84% He elevates it to an assumption then sees what will follow. 00:06:43.420 --> 00:06:45.970 align:middle line:84% And then it's from those starting postulates where, 00:06:45.970 --> 00:06:48.940 align:middle line:84% again, just in the opening few pages of this article, 00:06:48.940 --> 00:06:52.480 align:middle line:84% where he comes to what he calls the relativity of simultaneity. 00:06:52.480 --> 00:06:55.210 align:middle line:84% That events that we might consider 00:06:55.210 --> 00:06:57.070 align:middle line:84% to have happened at the same time, 00:06:57.070 --> 00:06:59.260 align:middle line:84% to have been simultaneous, will not 00:06:59.260 --> 00:07:03.220 align:middle line:84% appear to have been simultaneous to an observer who is otherwise 00:07:03.220 --> 00:07:05.890 align:middle line:84% fully legitimate, meaning all the laws of physics 00:07:05.890 --> 00:07:08.080 align:middle line:84% should hold perfectly valid for her, 00:07:08.080 --> 00:07:10.900 align:middle line:84% she just happens to be moving at some constant speed 00:07:10.900 --> 00:07:12.220 align:middle line:90% with respect to us. 00:07:12.220 --> 00:07:15.520 align:middle line:84% We will disagree on what counts as simultaneous. 00:07:15.520 --> 00:07:19.120 align:middle line:84% And from that, Einstein was convinced, 00:07:19.120 --> 00:07:22.510 align:middle line:84% would lead these other phenomena like length contraction 00:07:22.510 --> 00:07:26.080 align:middle line:84% and time dilation, not because of forces exerted by an ether, 00:07:26.080 --> 00:07:30.160 align:middle line:84% but because we disagree on when to implement our measurements, 00:07:30.160 --> 00:07:33.340 align:middle line:84% because we disagree on what counts as simultaneous events. 00:07:33.340 --> 00:07:34.750 align:middle line:90% And that comes for him-- 00:07:34.750 --> 00:07:37.240 align:middle line:84% from starting with kinematics, not dynamics. 00:07:37.240 --> 00:07:39.520 align:middle line:84% And again, that's the form of reasoning 00:07:39.520 --> 00:07:42.100 align:middle line:84% that Einstein puts into those opening 00:07:42.100 --> 00:07:44.300 align:middle line:90% pages of his paper from 1905. 00:07:44.300 --> 00:07:45.410 align:middle line:90% OK. 00:07:45.410 --> 00:07:47.997 align:middle line:84% So what were the early reactions to that body of work? 00:07:47.997 --> 00:07:49.580 align:middle line:84% Remember, it was published in, really, 00:07:49.580 --> 00:07:52.040 align:middle line:84% the leading, or certainly one of the leading journals 00:07:52.040 --> 00:07:54.650 align:middle line:84% in Western Europe for professional physicists. 00:07:54.650 --> 00:07:58.310 align:middle line:84% It was a perfect venue to try to get some attention. 00:07:58.310 --> 00:08:00.830 align:middle line:84% And yet, we now know historically the first reaction 00:08:00.830 --> 00:08:02.540 align:middle line:90% was no reaction at all. 00:08:02.540 --> 00:08:05.420 align:middle line:84% The baseline summary for today's class 00:08:05.420 --> 00:08:08.900 align:middle line:84% is no one paid attention or very few people paid attention. 00:08:08.900 --> 00:08:11.630 align:middle line:90% We now know Einstein's name. 00:08:11.630 --> 00:08:13.940 align:middle line:90% We might own Einstein's swag-- 00:08:13.940 --> 00:08:16.745 align:middle line:84% T-shirts and coffee mugs and bumper stickers, 00:08:16.745 --> 00:08:21.120 align:middle line:84% but that simply was not the case in 1905 or 1906 or 1907. 00:08:21.120 --> 00:08:22.910 align:middle line:90% It really was quite a delay. 00:08:22.910 --> 00:08:25.040 align:middle line:84% So at the time Einstein, as we saw, 00:08:25.040 --> 00:08:27.230 align:middle line:84% was this really little-known patent clerk. 00:08:27.230 --> 00:08:30.090 align:middle line:84% He was not employed as a research scientist. 00:08:30.090 --> 00:08:31.700 align:middle line:84% He was not teaching in a university. 00:08:31.700 --> 00:08:32.870 align:middle line:90% That had been his ambition. 00:08:32.870 --> 00:08:36.470 align:middle line:84% He just was not able to get that kind of job at first. 00:08:36.470 --> 00:08:40.070 align:middle line:84% He was publishing, before then, perfectly adequate, 00:08:40.070 --> 00:08:42.567 align:middle line:84% not very special articles on other topics 00:08:42.567 --> 00:08:43.400 align:middle line:90% in the main journal. 00:08:43.400 --> 00:08:45.350 align:middle line:84% He was building up a kind of portfolio 00:08:45.350 --> 00:08:49.100 align:middle line:84% of perfectly competent, not very earth-shattering work. 00:08:49.100 --> 00:08:52.475 align:middle line:84% So no one was hanging on, waiting at their mailbox 00:08:52.475 --> 00:08:54.350 align:middle line:84% to get the latest issue of the journal to see 00:08:54.350 --> 00:08:55.760 align:middle line:90% what Einstein had done lately. 00:08:55.760 --> 00:08:58.290 align:middle line:84% Very few people noticed anything at all. 00:08:58.290 --> 00:09:01.310 align:middle line:84% So this next point is one of my favorites. 00:09:01.310 --> 00:09:04.430 align:middle line:84% Not only did Einstein still not get the kind of job he had most 00:09:04.430 --> 00:09:07.190 align:middle line:84% wanted, an academic position, he didn't even 00:09:07.190 --> 00:09:09.770 align:middle line:84% get a raise or promotion at his current job. 00:09:09.770 --> 00:09:12.650 align:middle line:84% He remained a patent clerk third class 00:09:12.650 --> 00:09:15.320 align:middle line:84% even after publishing all of these articles even 00:09:15.320 --> 00:09:18.200 align:middle line:84% in the space of just one year within the leading physics 00:09:18.200 --> 00:09:18.753 align:middle line:90% journal. 00:09:18.753 --> 00:09:20.420 align:middle line:84% Not just the one we've looked at so far, 00:09:20.420 --> 00:09:22.250 align:middle line:90% even others we'll look at soon. 00:09:22.250 --> 00:09:24.790 align:middle line:84% He didn't even become patent clerk second class. 00:09:24.790 --> 00:09:28.850 align:middle line:84% This was really, really a complete shattering silence 00:09:28.850 --> 00:09:30.510 align:middle line:90% in response to his work. 00:09:30.510 --> 00:09:33.170 align:middle line:84% So one of his close friends, an exact contemporary 00:09:33.170 --> 00:09:35.840 align:middle line:84% of his, Max von Laue, whom he happened to meet, 00:09:35.840 --> 00:09:39.080 align:middle line:84% who was following a more traditional physics 00:09:39.080 --> 00:09:40.910 align:middle line:90% research and academic career-- 00:09:40.910 --> 00:09:43.100 align:middle line:84% von Laue was the same age as Einstein, 00:09:43.100 --> 00:09:46.400 align:middle line:84% but was making advances of the sort Einstein had hoped to make 00:09:46.400 --> 00:09:47.660 align:middle line:90% but was not yet making. 00:09:47.660 --> 00:09:51.230 align:middle line:84% Von Laue did his PhD quickly, he got a very prestigious 00:09:51.230 --> 00:09:52.880 align:middle line:84% postdoctoral position, he was soon 00:09:52.880 --> 00:09:55.820 align:middle line:84% teaching at a central research institute. 00:09:55.820 --> 00:09:59.720 align:middle line:84% Von Laue had befriended Einstein, they would talk, 00:09:59.720 --> 00:10:02.090 align:middle line:84% and because of his direct connection, 00:10:02.090 --> 00:10:04.880 align:middle line:84% Max von Laue and a very small number of others 00:10:04.880 --> 00:10:07.190 align:middle line:84% began trying to get a little more attention 00:10:07.190 --> 00:10:09.440 align:middle line:84% to Einstein's work on what we would now 00:10:09.440 --> 00:10:10.620 align:middle line:90% call special relativity. 00:10:10.620 --> 00:10:15.080 align:middle line:84% So for example, from his better-established academic 00:10:15.080 --> 00:10:17.720 align:middle line:84% position, von Laue wrote a review article 00:10:17.720 --> 00:10:19.140 align:middle line:84% that people did pay attention to. 00:10:19.140 --> 00:10:22.070 align:middle line:84% He was seen as a rising person to follow. 00:10:22.070 --> 00:10:24.830 align:middle line:84% He wrote an article fully four years later 00:10:24.830 --> 00:10:28.580 align:middle line:84% in 1909 which helped to bring Einstein's work to others' 00:10:28.580 --> 00:10:31.350 align:middle line:84% attention, Einstein's work and that work of others as well. 00:10:31.350 --> 00:10:34.520 align:middle line:84% So a few years later, some more people 00:10:34.520 --> 00:10:36.470 align:middle line:84% begin to pay attention not because they 00:10:36.470 --> 00:10:39.530 align:middle line:84% were following Einstein, per se, but a little bit of a kind 00:10:39.530 --> 00:10:41.750 align:middle line:90% of percolation starts. 00:10:41.750 --> 00:10:44.220 align:middle line:84% So reaction one is really no reaction at all. 00:10:44.220 --> 00:10:47.480 align:middle line:84% The second reaction didn't make Einstein much happier. 00:10:47.480 --> 00:10:50.330 align:middle line:84% So when people did notice the work, the minority, 00:10:50.330 --> 00:10:53.540 align:middle line:84% the exceptions to that rule who paid any attention at all, 00:10:53.540 --> 00:10:57.110 align:middle line:84% they tended to read Einstein's paper, the one that for all 00:10:57.110 --> 00:11:02.240 align:middle line:84% immersed in now, as a clever rederivation of what was, 00:11:02.240 --> 00:11:04.130 align:middle line:90% by then old news. 00:11:04.130 --> 00:11:07.010 align:middle line:84% They saw this as a really kind of smart way 00:11:07.010 --> 00:11:10.580 align:middle line:84% to get back to the results that leading physicists, 00:11:10.580 --> 00:11:14.330 align:middle line:84% like Hendrik Lorentz and Henri Poincare in Paris and others, 00:11:14.330 --> 00:11:18.410 align:middle line:84% had been finding for a decade since the 1890s. 00:11:18.410 --> 00:11:21.890 align:middle line:84% Remember, we even-- we saw this in our own class discussions 00:11:21.890 --> 00:11:23.930 align:middle line:90% a few sessions ago. 00:11:23.930 --> 00:11:26.150 align:middle line:84% Hendrik Lorentz had already published 00:11:26.150 --> 00:11:27.260 align:middle line:90% on length contraction. 00:11:27.260 --> 00:11:30.830 align:middle line:84% He was directly responding to the Michelson-Morley experiment 00:11:30.830 --> 00:11:32.030 align:middle line:90% in Lorentz's case. 00:11:32.030 --> 00:11:34.850 align:middle line:84% He hadn't really argued about there should be a contraction. 00:11:34.850 --> 00:11:37.910 align:middle line:84% He had derived quantitatively an expression involving 00:11:37.910 --> 00:11:40.730 align:middle line:84% this factor gamma that we've already seen many times, 00:11:40.730 --> 00:11:43.690 align:middle line:84% 1 over the square root of 1 minus v over c squared. 00:11:43.690 --> 00:11:46.150 align:middle line:84% That was the exact same quantitative result 00:11:46.150 --> 00:11:48.490 align:middle line:84% to which Einstein had arrived by very different forms 00:11:48.490 --> 00:11:49.360 align:middle line:90% of reasoning. 00:11:49.360 --> 00:11:53.770 align:middle line:84% Lorenz was a very senior, very established, very productive 00:11:53.770 --> 00:11:55.060 align:middle line:90% mathematical physicist. 00:11:55.060 --> 00:11:57.910 align:middle line:84% People really did want to see what was Lorenz's latest work. 00:11:57.910 --> 00:12:00.820 align:middle line:84% They really would hang on his latest articles. 00:12:00.820 --> 00:12:03.490 align:middle line:84% So the small number, the exceptions 00:12:03.490 --> 00:12:05.530 align:middle line:84% who noticed Einstein's work in 1905 00:12:05.530 --> 00:12:07.930 align:middle line:90% said, oh, I've seen that before. 00:12:07.930 --> 00:12:10.090 align:middle line:84% It's a clever way of getting back 00:12:10.090 --> 00:12:12.950 align:middle line:84% to what was by then 10-year-old results. 00:12:12.950 --> 00:12:16.690 align:middle line:84% And so if anyone gave Einstein credit in the early years, 00:12:16.690 --> 00:12:18.880 align:middle line:90% it was as like second fiddle. 00:12:18.880 --> 00:12:22.300 align:middle line:84% They would refer sometimes to Lorenz's theory, Lorenz's 00:12:22.300 --> 00:12:23.890 align:middle line:84% electrodynamics, or a few of them 00:12:23.890 --> 00:12:26.080 align:middle line:84% would refer to the Lorenz-Einstein theory. 00:12:26.080 --> 00:12:28.730 align:middle line:84% No one referred to the Einstein-Lorenz theory. 00:12:28.730 --> 00:12:31.780 align:middle line:84% So the second reaction didn't make Einstein much happier. 00:12:31.780 --> 00:12:33.760 align:middle line:84% Either people ignored it or they somehow, 00:12:33.760 --> 00:12:36.040 align:middle line:84% as far as he was concerned, misunderstood 00:12:36.040 --> 00:12:41.390 align:middle line:84% this as a trivial reworking of more established results. 00:12:41.390 --> 00:12:44.630 align:middle line:84% So even when people gave Einstein's work any attention 00:12:44.630 --> 00:12:48.650 align:middle line:84% within this rubric of the framework that Lorenz had been 00:12:48.650 --> 00:12:50.870 align:middle line:84% establishing throughout the 1890s, 00:12:50.870 --> 00:12:54.950 align:middle line:84% few people seem to notice that the conceptual underpinning was 00:12:54.950 --> 00:12:56.510 align:middle line:90% really quite distinct. 00:12:56.510 --> 00:12:58.500 align:middle line:84% Some of the equations were identical, 00:12:58.500 --> 00:13:00.740 align:middle line:84% the same factor of gamma, the same coordinate 00:13:00.740 --> 00:13:02.400 align:middle line:90% transformations and so on. 00:13:02.400 --> 00:13:04.640 align:middle line:84% And yet, they seem to be based, as we would now 00:13:04.640 --> 00:13:07.160 align:middle line:84% see with hindsight, on very, very 00:13:07.160 --> 00:13:09.800 align:middle line:90% different starting assumptions. 00:13:09.800 --> 00:13:12.380 align:middle line:84% Lorenz's work was all about the ether. 00:13:12.380 --> 00:13:13.910 align:middle line:84% The ether wasn't only central, it 00:13:13.910 --> 00:13:16.100 align:middle line:84% was exerting forces, these dynamics 00:13:16.100 --> 00:13:17.540 align:middle line:90% according to Lorenz's view. 00:13:17.540 --> 00:13:21.590 align:middle line:84% It was literally squeezing the atoms and molecules in the arm 00:13:21.590 --> 00:13:22.880 align:middle line:90% of Michelson's device. 00:13:22.880 --> 00:13:26.120 align:middle line:84% It was all about the elasticity and the forces 00:13:26.120 --> 00:13:28.280 align:middle line:90% exerted by a physical ether. 00:13:28.280 --> 00:13:30.470 align:middle line:84% And as we've seen in Einstein's case, 00:13:30.470 --> 00:13:32.240 align:middle line:84% in the opening few paragraphs, he's 00:13:32.240 --> 00:13:34.220 align:middle line:84% dismissed the ether as superfluous. 00:13:34.220 --> 00:13:36.110 align:middle line:90% His work is not about the ether. 00:13:36.110 --> 00:13:38.875 align:middle line:84% He says the ether was a century-long distraction, which 00:13:38.875 --> 00:13:41.000 align:middle line:84% is a very different starting point even though they 00:13:41.000 --> 00:13:43.530 align:middle line:84% both arrive at some very similar equations. 00:13:43.530 --> 00:13:45.410 align:middle line:84% So even when people gave it attention, 00:13:45.410 --> 00:13:48.170 align:middle line:84% they misread it or they read it in ways different from how 00:13:48.170 --> 00:13:51.290 align:middle line:90% Einstein probably had hoped. 00:13:51.290 --> 00:13:54.530 align:middle line:84% And again, this lasts not just until 1906 or '07. 00:13:54.530 --> 00:13:58.910 align:middle line:84% Even after 1909, review articles by people like Max von Laue. 00:13:58.910 --> 00:14:02.077 align:middle line:84% Here's a great example that I take from Andy Warwick's work. 00:14:02.077 --> 00:14:04.160 align:middle line:84% We had some more of Andy's work to read for today, 00:14:04.160 --> 00:14:06.620 align:middle line:84% and I'll talk about that later in today's class. 00:14:06.620 --> 00:14:08.840 align:middle line:84% But he has this great example in his article. 00:14:08.840 --> 00:14:11.130 align:middle line:90% That even as late as 1913-- 00:14:11.130 --> 00:14:13.310 align:middle line:84% now we're talking nearly a decade later. 00:14:13.310 --> 00:14:15.200 align:middle line:90% So fully eight years later. 00:14:15.200 --> 00:14:17.570 align:middle line:84% A leading British physicist had written 00:14:17.570 --> 00:14:20.300 align:middle line:84% about what he called "The abstruse conceptions" 00:14:20.300 --> 00:14:23.240 align:middle line:84% of Einstein's 1905 paper, which were, quote, 00:14:23.240 --> 00:14:25.400 align:middle line:84% "most foreign to our habits of thought," 00:14:25.400 --> 00:14:27.980 align:middle line:84% and "as yet scarcely anyone in this country," 00:14:27.980 --> 00:14:30.410 align:middle line:84% by which he means Britain, "professed to understand 00:14:30.410 --> 00:14:31.670 align:middle line:90% or at least to appreciate." 00:14:31.670 --> 00:14:34.070 align:middle line:84% So even the people who knew there was this person 00:14:34.070 --> 00:14:36.560 align:middle line:84% named Einstein who worked on the electrodynamics of moving 00:14:36.560 --> 00:14:39.470 align:middle line:84% bodies, this was not kind of commanding attention in the way 00:14:39.470 --> 00:14:42.200 align:middle line:84% that we might expect because we know how the story ends. 00:14:42.200 --> 00:14:45.260 align:middle line:84% We know what Einstein would go on to be recognized for. 00:14:45.260 --> 00:14:48.620 align:middle line:84% It was a pretty slow transformation. 00:14:48.620 --> 00:14:51.660 align:middle line:84% And we see this reflected in Einstein's own career as well. 00:14:51.660 --> 00:14:54.620 align:middle line:84% So as I mentioned, one of my favorite factoids 00:14:54.620 --> 00:14:57.082 align:middle line:84% for the entire class today, Einstein didn't-- not only 00:14:57.082 --> 00:14:59.540 align:middle line:84% didn't leave the patent office, he didn't even get promoted 00:14:59.540 --> 00:15:01.940 align:middle line:84% within the patent office in the light of this quite 00:15:01.940 --> 00:15:04.370 align:middle line:84% extraordinary work, his first break, 00:15:04.370 --> 00:15:08.062 align:middle line:84% the first opportunity for the kind of career we know that he 00:15:08.062 --> 00:15:10.520 align:middle line:84% really wanted-- we know from his personal letters that have 00:15:10.520 --> 00:15:14.720 align:middle line:84% survived and are available, he had been hoping for an academic 00:15:14.720 --> 00:15:17.000 align:middle line:84% position from his early, early days. 00:15:17.000 --> 00:15:20.780 align:middle line:84% His first big break along that new career path 00:15:20.780 --> 00:15:24.020 align:middle line:84% comes actually in 1909, around the time 00:15:24.020 --> 00:15:26.630 align:middle line:84% that von Laue was writing that review article and others, 00:15:26.630 --> 00:15:28.460 align:middle line:84% where he was offered an assistant professor 00:15:28.460 --> 00:15:30.590 align:middle line:84% position in Zurich, which was actually very exciting. 00:15:30.590 --> 00:15:32.298 align:middle line:84% That's where he had done his own school-- 00:15:32.298 --> 00:15:34.040 align:middle line:90% his own university training. 00:15:34.040 --> 00:15:38.060 align:middle line:84% Then two years later, he's given a promotion, an offer 00:15:38.060 --> 00:15:40.160 align:middle line:90% to move now to Prague. 00:15:40.160 --> 00:15:42.530 align:middle line:84% He spends about one year there, then 00:15:42.530 --> 00:15:45.830 align:middle line:84% he's hired back at a different institution, back in Zurich. 00:15:45.830 --> 00:15:51.050 align:middle line:84% And then finally, in the spring of 1914, in April or May, 00:15:51.050 --> 00:15:53.300 align:middle line:84% he's invited to join the very, very 00:15:53.300 --> 00:15:56.030 align:middle line:84% prestigious Prussian Academy of Sciences in Berlin. 00:15:56.030 --> 00:15:59.300 align:middle line:84% So roughly a decade later, he's promoted 00:15:59.300 --> 00:16:03.230 align:middle line:84% to the kind of position he'd been really angling for all 00:16:03.230 --> 00:16:05.635 align:middle line:84% along, and it comes not one year, not 00:16:05.635 --> 00:16:07.010 align:middle line:84% two years, not three years later, 00:16:07.010 --> 00:16:08.390 align:middle line:90% but nearly a decade later. 00:16:08.390 --> 00:16:11.210 align:middle line:84% And after a lot more work has come out, not 00:16:11.210 --> 00:16:13.580 align:middle line:84% just on the basis of this 1905 work. 00:16:13.580 --> 00:16:17.570 align:middle line:84% So has a slowly accelerating academic career 00:16:17.570 --> 00:16:21.090 align:middle line:84% starting from this very unusual starting point. 00:16:21.090 --> 00:16:22.170 align:middle line:90% So I'll pause there. 00:16:22.170 --> 00:16:24.965 align:middle line:84% Any questions on that early discussion 00:16:24.965 --> 00:16:26.465 align:middle line:84% of the reception of Einstein's work? 00:16:26.465 --> 00:16:33.050 align:middle line:90% 00:16:33.050 --> 00:16:35.840 align:middle line:84% This is a good moment to remind you, try not to annoy 00:16:35.840 --> 00:16:37.198 align:middle line:90% all your professors in college. 00:16:37.198 --> 00:16:38.990 align:middle line:84% You'll get better letters of recommendation 00:16:38.990 --> 00:16:40.250 align:middle line:84% that will help you on the job market. 00:16:40.250 --> 00:16:41.960 align:middle line:84% That was one of many things Einstein 00:16:41.960 --> 00:16:44.870 align:middle line:84% had done so poorly because he was kind of a dummy. 00:16:44.870 --> 00:16:46.940 align:middle line:84% Einstein, as I like to say, was no Einstein when 00:16:46.940 --> 00:16:48.148 align:middle line:90% it came to his own schooling. 00:16:48.148 --> 00:16:51.320 align:middle line:84% So that's life lesson number one for 8225. 00:16:51.320 --> 00:16:52.370 align:middle line:90% Write that down. 00:16:52.370 --> 00:16:54.560 align:middle line:84% Any other questions about Einstein's-- the early 00:16:54.560 --> 00:16:57.290 align:middle line:84% reception of either his own career or this body of work? 00:16:57.290 --> 00:17:01.070 align:middle line:90% If not, I'll gladly press on. 00:17:01.070 --> 00:17:01.570 align:middle line:90% OK. 00:17:01.570 --> 00:17:03.362 align:middle line:84% If questions come up, put them in the chat, 00:17:03.362 --> 00:17:05.468 align:middle line:84% we'll have more time to talk about them. 00:17:05.468 --> 00:17:06.760 align:middle line:90% But I'll go back to the slides. 00:17:06.760 --> 00:17:08.470 align:middle line:84% We'll talk about now this next part. 00:17:08.470 --> 00:17:10.480 align:middle line:90% We can dig in a bit more to-- 00:17:10.480 --> 00:17:13.300 align:middle line:84% an example of a bit more significant engagement, 00:17:13.300 --> 00:17:17.170 align:middle line:84% a stronger reaction and response to Einstein's work. 00:17:17.170 --> 00:17:19.900 align:middle line:84% And that came actually somewhat surprisingly 00:17:19.900 --> 00:17:22.359 align:middle line:84% from one of Einstein's former mathematics 00:17:22.359 --> 00:17:24.010 align:middle line:90% teachers, Herman Minkowski. 00:17:24.010 --> 00:17:27.670 align:middle line:84% So Minkowski was a very elite established 00:17:27.670 --> 00:17:29.240 align:middle line:90% professional mathematician. 00:17:29.240 --> 00:17:32.200 align:middle line:90% He was a professor at the ETH-- 00:17:32.200 --> 00:17:34.390 align:middle line:84% that's the university that Einstein 00:17:34.390 --> 00:17:38.050 align:middle line:84% had worked so hard to get into, the Eidgenossische Technische 00:17:38.050 --> 00:17:40.780 align:middle line:84% Hochschule, which I never tire of trying to pronounce, 00:17:40.780 --> 00:17:42.040 align:middle line:90% we can just call it ETH. 00:17:42.040 --> 00:17:45.880 align:middle line:84% That was the Swiss Federal Polyatomic Institute in Zurich, 00:17:45.880 --> 00:17:47.800 align:middle line:90% kind of like MIT. 00:17:47.800 --> 00:17:51.160 align:middle line:84% Minkowski was a mathematics professor, not a physicist. 00:17:51.160 --> 00:17:53.920 align:middle line:84% And through a kind of roundabout way, 00:17:53.920 --> 00:17:57.100 align:middle line:84% Einstein's work was brought back to Minkowski's attention 00:17:57.100 --> 00:18:00.100 align:middle line:84% a few years later post-publication. 00:18:00.100 --> 00:18:03.260 align:middle line:84% So it turns out, Minkowski was certainly not looking 00:18:03.260 --> 00:18:04.950 align:middle line:84% for Einstein's work, and we have, again, 00:18:04.950 --> 00:18:07.640 align:middle line:84% some wonderful, juicy, colorful letters 00:18:07.640 --> 00:18:09.890 align:middle line:84% that survive of Minkowski basically 00:18:09.890 --> 00:18:12.450 align:middle line:84% talking behind Einstein's back and vice versa. 00:18:12.450 --> 00:18:15.710 align:middle line:84% So we know that when Einstein was a university student, 00:18:15.710 --> 00:18:17.810 align:middle line:84% he'd worked so hard to get into ETH. 00:18:17.810 --> 00:18:18.780 align:middle line:90% What did he do? 00:18:18.780 --> 00:18:21.920 align:middle line:84% He began cutting classes, especially classes of faculty 00:18:21.920 --> 00:18:24.920 align:middle line:84% for whom he had little respect, and that happens 00:18:24.920 --> 00:18:26.390 align:middle line:90% to include Herman Minkowski. 00:18:26.390 --> 00:18:29.240 align:middle line:84% So even though he was registered and took classes for credit 00:18:29.240 --> 00:18:31.310 align:middle line:84% from Minkowski in the Math Department, 00:18:31.310 --> 00:18:34.550 align:middle line:84% he rarely showed up, and Minkowski noticed that. 00:18:34.550 --> 00:18:37.340 align:middle line:84% So Einstein used to borrow notes from several friends, 00:18:37.340 --> 00:18:39.740 align:middle line:84% from his, at the time, girlfriend, who became 00:18:39.740 --> 00:18:41.360 align:middle line:90% his first wife, Mileva Maric. 00:18:41.360 --> 00:18:43.725 align:middle line:84% She was also a physics student at the ETH 00:18:43.725 --> 00:18:46.100 align:middle line:84% and doing actually quite well in her mathematics classes. 00:18:46.100 --> 00:18:48.980 align:middle line:84% She would very dutifully attend, take very careful notes, 00:18:48.980 --> 00:18:51.020 align:middle line:84% share them with her good-for-nothing boyfriend 00:18:51.020 --> 00:18:51.860 align:middle line:90% Albert Einstein. 00:18:51.860 --> 00:18:55.170 align:middle line:84% He would cram before the exam and kind of do OK. 00:18:55.170 --> 00:18:56.870 align:middle line:84% Another one of Einstein's close buddies 00:18:56.870 --> 00:18:59.990 align:middle line:84% from undergraduate days, Marcel Grossmann, did the same thing. 00:18:59.990 --> 00:19:03.260 align:middle line:84% Grossmann went on to a career as a mathematician himself. 00:19:03.260 --> 00:19:05.480 align:middle line:84% So Einstein would basically cut classes 00:19:05.480 --> 00:19:09.410 align:middle line:84% from people like Minkowski and then catch up and just 00:19:09.410 --> 00:19:10.100 align:middle line:90% skate through. 00:19:10.100 --> 00:19:12.890 align:middle line:84% He would just like barely pass by cramming the night before. 00:19:12.890 --> 00:19:14.970 align:middle line:84% Don't do that on paper 1, please. 00:19:14.970 --> 00:19:18.230 align:middle line:84% So a friend of Einstein's encouraged 00:19:18.230 --> 00:19:21.080 align:middle line:84% the professor, Minkowski, to read this 1905 00:19:21.080 --> 00:19:23.750 align:middle line:84% paper a few years after it'd come out. 00:19:23.750 --> 00:19:25.460 align:middle line:84% And Minkowski wrote back, I really 00:19:25.460 --> 00:19:27.650 align:middle line:84% wouldn't have thought Einstein capable of that. 00:19:27.650 --> 00:19:30.320 align:middle line:84% He wrote-- Minkowski wrote to another colleague, 00:19:30.320 --> 00:19:33.050 align:middle line:84% Einstein's paper came as a tremendous surprise, 00:19:33.050 --> 00:19:36.380 align:middle line:84% because in his student days, Einstein had been a lazy dog. 00:19:36.380 --> 00:19:38.600 align:middle line:84% He never bothered about mathematics at all. 00:19:38.600 --> 00:19:41.390 align:middle line:84% So much for Minkowski having a high opinion 00:19:41.390 --> 00:19:43.190 align:middle line:90% of young Albert Einstein. 00:19:43.190 --> 00:19:45.950 align:middle line:84% So once Minkowski was convinced to even look 00:19:45.950 --> 00:19:48.920 align:middle line:84% at this again a few years post-publication, 00:19:48.920 --> 00:19:51.290 align:middle line:84% he quickly became convinced that Einstein was still 00:19:51.290 --> 00:19:54.200 align:middle line:84% a lazy dog or at least was still making things unnecessarily 00:19:54.200 --> 00:19:55.260 align:middle line:90% complicated. 00:19:55.260 --> 00:19:58.340 align:middle line:84% So Minkowski thought Einstein had a few interesting ideas 00:19:58.340 --> 00:20:02.330 align:middle line:84% in this paper from 1905, but had really messed things up as far 00:20:02.330 --> 00:20:03.740 align:middle line:90% as Minkowski was concerned. 00:20:03.740 --> 00:20:07.280 align:middle line:84% That he had really missed the main point of his own work. 00:20:07.280 --> 00:20:09.710 align:middle line:84% Minkowski was convinced that Einstein's understanding 00:20:09.710 --> 00:20:13.110 align:middle line:84% of Einstein's work was not the best understanding at all. 00:20:13.110 --> 00:20:16.370 align:middle line:84% So in fact, Minkowski set about reformulating this work 00:20:16.370 --> 00:20:18.980 align:middle line:84% in his own way in around 1907, it was actually 00:20:18.980 --> 00:20:21.650 align:middle line:90% published posthumously in 1908. 00:20:21.650 --> 00:20:24.410 align:middle line:84% Minkowski died rather young, so he gave this famous lecture 00:20:24.410 --> 00:20:28.200 align:middle line:84% and it was published soon after he passed away. 00:20:28.200 --> 00:20:30.530 align:middle line:84% Who was this person, Herman Minkowski? 00:20:30.530 --> 00:20:33.830 align:middle line:84% He wasn't only a mathematician at the ETH in Zurich, 00:20:33.830 --> 00:20:35.870 align:middle line:84% he was, in particular, a geometer. 00:20:35.870 --> 00:20:39.500 align:middle line:84% His specialty within mathematics was pure geometry. 00:20:39.500 --> 00:20:43.250 align:middle line:84% In fact, he was like an evangelist for geometry. 00:20:43.250 --> 00:20:46.710 align:middle line:84% He had written a book, first published in 1896-- 00:20:46.710 --> 00:20:48.410 align:middle line:84% the cover here is from a later edition-- 00:20:48.410 --> 00:20:50.390 align:middle line:90% called the Geometry of Numbers. 00:20:50.390 --> 00:20:52.730 align:middle line:84% He actually wanted to remake abstract fields 00:20:52.730 --> 00:20:55.040 align:middle line:84% of pure mathematics like number theory, 00:20:55.040 --> 00:20:57.360 align:middle line:84% make those part of geometry as well. 00:20:57.360 --> 00:20:59.720 align:middle line:84% So he wasn't only an expert in geometry, 00:20:59.720 --> 00:21:02.730 align:middle line:84% he thought geometry would unlock the secrets for all of math 00:21:02.730 --> 00:21:03.980 align:middle line:90% and that's all he cared about. 00:21:03.980 --> 00:21:05.510 align:middle line:90% That was the key to everything. 00:21:05.510 --> 00:21:09.160 align:middle line:84% So he really was a geometer all the way through. 00:21:09.160 --> 00:21:12.000 align:middle line:84% So when he then later turned to Einstein's work 00:21:12.000 --> 00:21:14.820 align:middle line:84% about two years after it had been published, 00:21:14.820 --> 00:21:16.860 align:middle line:90% he did so as a geometer. 00:21:16.860 --> 00:21:19.980 align:middle line:84% Not as a physicist, not as a philosopher interested 00:21:19.980 --> 00:21:22.470 align:middle line:90% in math or positivism. 00:21:22.470 --> 00:21:24.780 align:middle line:84% To Minkowski, Einstein's work held 00:21:24.780 --> 00:21:28.590 align:middle line:84% lessons that were best understood through using 00:21:28.590 --> 00:21:30.460 align:middle line:90% the tools of geometry. 00:21:30.460 --> 00:21:32.400 align:middle line:84% So one of the first things Minkowski did 00:21:32.400 --> 00:21:34.470 align:middle line:84% is something we all take for granted today. 00:21:34.470 --> 00:21:36.460 align:middle line:84% Many of you have probably seen this before. 00:21:36.460 --> 00:21:39.330 align:middle line:84% We now will call these things space-time diagrams, 00:21:39.330 --> 00:21:41.537 align:middle line:84% or often we'll call them Minkowski diagrams. 00:21:41.537 --> 00:21:43.620 align:middle line:84% These were one of the things that Herman Minkowski 00:21:43.620 --> 00:21:47.550 align:middle line:84% introduced once he grudgingly began to pay attention to this 00:21:47.550 --> 00:21:48.930 align:middle line:90% paper of Albert Einstein's. 00:21:48.930 --> 00:21:52.530 align:middle line:84% So what are we going to do with these space-time diagrams? 00:21:52.530 --> 00:21:54.810 align:middle line:84% To make things simple, we'll consider motion 00:21:54.810 --> 00:21:57.850 align:middle line:84% in only one direction of space, let's say the x-axis. 00:21:57.850 --> 00:22:01.350 align:middle line:84% So quite typically, the convention very quickly became, 00:22:01.350 --> 00:22:05.700 align:middle line:84% we'll measure locations in space along the horizontal axis 00:22:05.700 --> 00:22:07.020 align:middle line:90% or the x-axis. 00:22:07.020 --> 00:22:11.470 align:middle line:84% And we'll measure changes in time along the vertical axis. 00:22:11.470 --> 00:22:14.440 align:middle line:84% So now we have a two-dimensional plot to measure-- 00:22:14.440 --> 00:22:17.200 align:middle line:84% with which to make sense of motion in space and time. 00:22:17.200 --> 00:22:19.600 align:middle line:90% These are space-time diagrams. 00:22:19.600 --> 00:22:22.480 align:middle line:84% What are the next things that became very, very common very 00:22:22.480 --> 00:22:24.880 align:middle line:84% quickly with these kinds of graphs 00:22:24.880 --> 00:22:28.540 align:middle line:84% was to use very convenient coordinates, coordinates 00:22:28.540 --> 00:22:31.930 align:middle line:84% in which light waves would travel one unit of space, 00:22:31.930 --> 00:22:35.350 align:middle line:84% one tick along the x-axis, for every one tick, 00:22:35.350 --> 00:22:37.400 align:middle line:90% for every one unit of time. 00:22:37.400 --> 00:22:39.460 align:middle line:84% So for example, if we're going to measure time 00:22:39.460 --> 00:22:43.310 align:middle line:84% in seconds, a very common unit with which to measure time, 00:22:43.310 --> 00:22:45.850 align:middle line:84% then we better not measure spatial distances 00:22:45.850 --> 00:22:50.140 align:middle line:84% in either meters or feet or kilometers or even parsecs. 00:22:50.140 --> 00:22:52.490 align:middle line:84% We're going to measure them in light-seconds, 00:22:52.490 --> 00:22:55.760 align:middle line:84% which is the distance light will travel in one second. 00:22:55.760 --> 00:22:58.000 align:middle line:84% So we're going to measure the distances 00:22:58.000 --> 00:22:59.920 align:middle line:84% in units such that light will travel 00:22:59.920 --> 00:23:03.075 align:middle line:84% to one of those spatial units in one unit of time. 00:23:03.075 --> 00:23:04.700 align:middle line:84% For example, seconds and light-seconds. 00:23:04.700 --> 00:23:06.283 align:middle line:84% So what we're really doing, in effect, 00:23:06.283 --> 00:23:08.530 align:middle line:84% is scaling the speed of light to be 1. 00:23:08.530 --> 00:23:12.280 align:middle line:84% And therefore, when we go to plot the motion of light rays 00:23:12.280 --> 00:23:15.340 align:middle line:84% or light waves on these space-time diagrams, 00:23:15.340 --> 00:23:18.910 align:middle line:84% they'll follow these very simple 45-degree diagonals. 00:23:18.910 --> 00:23:22.720 align:middle line:84% Their slope is fixed to be always inclined at 45 degrees 00:23:22.720 --> 00:23:25.240 align:middle line:84% because we've chosen our coordinates such 00:23:25.240 --> 00:23:27.940 align:middle line:84% that they will always traverse one unit of space 00:23:27.940 --> 00:23:29.090 align:middle line:90% in one unit of time. 00:23:29.090 --> 00:23:31.630 align:middle line:84% Now why will they always travel at that slope? 00:23:31.630 --> 00:23:34.570 align:middle line:84% Because if we take Einstein's second postulate seriously, 00:23:34.570 --> 00:23:38.038 align:middle line:84% light will always be traveling at that single universal speed, 00:23:38.038 --> 00:23:38.830 align:middle line:90% the speed of light. 00:23:38.830 --> 00:23:41.200 align:middle line:84% Therefore, they'll have a single fixed slope. 00:23:41.200 --> 00:23:43.210 align:middle line:84% They could travel off to the left, 00:23:43.210 --> 00:23:45.070 align:middle line:84% and therefore, then we'd have a worldline 00:23:45.070 --> 00:23:48.887 align:middle line:84% like this inclined at 45 degrees pointing off to the left. 00:23:48.887 --> 00:23:50.470 align:middle line:84% They can travel to the right, but they 00:23:50.470 --> 00:23:51.610 align:middle line:90% can't change the slope. 00:23:51.610 --> 00:23:54.950 align:middle line:84% That's the impact of Einstein's second postulate. 00:23:54.950 --> 00:23:55.450 align:middle line:90% OK. 00:23:55.450 --> 00:23:56.560 align:middle line:90% So far, so good. 00:23:56.560 --> 00:23:59.830 align:middle line:84% We can do things like we can plot more than just the path 00:23:59.830 --> 00:24:01.030 align:middle line:90% of light. 00:24:01.030 --> 00:24:02.932 align:middle line:90% Here's this-- this is real data. 00:24:02.932 --> 00:24:05.140 align:middle line:84% This is how I spend my time these days in quarantine. 00:24:05.140 --> 00:24:06.280 align:middle line:90% I don't go anywhere. 00:24:06.280 --> 00:24:11.620 align:middle line:84% So my position, the plot of my motion through space and time 00:24:11.620 --> 00:24:13.510 align:middle line:90% is really easy now. 00:24:13.510 --> 00:24:17.170 align:middle line:84% I just sit in one location in space, like this chair, 00:24:17.170 --> 00:24:18.010 align:middle line:90% and I don't move. 00:24:18.010 --> 00:24:20.650 align:middle line:84% So all that happens is time ticks inexorably 00:24:20.650 --> 00:24:23.350 align:middle line:84% by, I just get older and older and older, 00:24:23.350 --> 00:24:25.450 align:middle line:84% but I'm not moving around, or barely. 00:24:25.450 --> 00:24:28.120 align:middle line:84% So we can plot what's called my worldline. 00:24:28.120 --> 00:24:30.280 align:middle line:84% That's the vocabulary that gets used now 00:24:30.280 --> 00:24:32.860 align:middle line:84% with these space-time diagrams, dating really back 00:24:32.860 --> 00:24:35.650 align:middle line:90% to the era of Herman Minkowski. 00:24:35.650 --> 00:24:38.440 align:middle line:84% So for any object, we can chart its motion 00:24:38.440 --> 00:24:40.540 align:middle line:84% through space and time in one of these plots. 00:24:40.540 --> 00:24:42.460 align:middle line:84% If it's not changing its location, 00:24:42.460 --> 00:24:45.122 align:middle line:84% but only moving through time, it will have a straight line 00:24:45.122 --> 00:24:46.330 align:middle line:90% heading straight up the page. 00:24:46.330 --> 00:24:47.860 align:middle line:84% It will move-- if it's light, it's 00:24:47.860 --> 00:24:51.460 align:middle line:84% going to move at this fixed inclination or slope. 00:24:51.460 --> 00:24:54.580 align:middle line:84% And of course, if my family forces me to get off the chair 00:24:54.580 --> 00:24:57.550 align:middle line:84% and go out for a jog, then I'll have a more complicated 00:24:57.550 --> 00:24:58.090 align:middle line:90% worldline. 00:24:58.090 --> 00:25:01.270 align:middle line:84% I'll be moving, I'll be changing my location over time, 00:25:01.270 --> 00:25:04.400 align:middle line:84% not nearly as fast as light, not nearly as fast as shown here. 00:25:04.400 --> 00:25:06.040 align:middle line:90% This is like supersonic jetpack. 00:25:06.040 --> 00:25:08.950 align:middle line:84% But I will sometimes be moving to the right at some speed. 00:25:08.950 --> 00:25:11.440 align:middle line:84% I might change directions later and move to the left. 00:25:11.440 --> 00:25:13.930 align:middle line:84% But whatever I do, I'll be moving at a speed slower 00:25:13.930 --> 00:25:15.670 align:middle line:90% than light, that's for sure. 00:25:15.670 --> 00:25:17.440 align:middle line:84% And therefore, the slope of my worldline 00:25:17.440 --> 00:25:20.620 align:middle line:84% is limited to never be as sharp as inclined 00:25:20.620 --> 00:25:22.370 align:middle line:90% as a 45-degree line. 00:25:22.370 --> 00:25:25.420 align:middle line:84% So I might have a nontrivial worldline, 00:25:25.420 --> 00:25:28.090 align:middle line:84% but it's a way, nonetheless, to chart my motion through space 00:25:28.090 --> 00:25:28.758 align:middle line:90% and time. 00:25:28.758 --> 00:25:30.550 align:middle line:84% Many of you have probably seen that before. 00:25:30.550 --> 00:25:33.610 align:middle line:84% This comes from the geometer Herman Minkowski 00:25:33.610 --> 00:25:35.680 align:middle line:84% trying to make sense of Einstein's work. 00:25:35.680 --> 00:25:36.250 align:middle line:90% OK. 00:25:36.250 --> 00:25:38.200 align:middle line:84% Now let's go back to one of the scenarios 00:25:38.200 --> 00:25:41.230 align:middle line:84% that Einstein introduced in those opening pages, the part 00:25:41.230 --> 00:25:44.860 align:middle line:84% that you have for your assignment of his 1905 paper. 00:25:44.860 --> 00:25:46.840 align:middle line:84% Let's imagine that I'm standing still-- 00:25:46.840 --> 00:25:50.020 align:middle line:84% so I'm back to my very trivial straight-up-the page worldline, 00:25:50.020 --> 00:25:51.370 align:middle line:90% I'm back in my chair. 00:25:51.370 --> 00:25:54.280 align:middle line:84% And I'm standing exactly equal distance 00:25:54.280 --> 00:25:57.640 align:middle line:84% from two colleagues who are at locations A and B. 00:25:57.640 --> 00:26:01.180 align:middle line:84% So I'm an equal distance along the x-axis 00:26:01.180 --> 00:26:02.500 align:middle line:90% from each of my colleagues. 00:26:02.500 --> 00:26:05.230 align:middle line:84% I've asked them to please shine their lanterns at me 00:26:05.230 --> 00:26:06.670 align:middle line:90% at the same time. 00:26:06.670 --> 00:26:08.260 align:middle line:84% If they've really done that, then I'll 00:26:08.260 --> 00:26:11.530 align:middle line:84% receive the light waves from each of them at the same time. 00:26:11.530 --> 00:26:14.320 align:middle line:84% The light wave traveling from B toward me 00:26:14.320 --> 00:26:16.780 align:middle line:84% and the light wave traveling from A toward me 00:26:16.780 --> 00:26:20.050 align:middle line:84% will arrive at my location at the same time-- 00:26:20.050 --> 00:26:20.650 align:middle line:90% sorry. 00:26:20.650 --> 00:26:24.040 align:middle line:84% And if so, then I know that A and B emitted their light waves 00:26:24.040 --> 00:26:27.370 align:middle line:84% simultaneously because the speed of light is fixed. 00:26:27.370 --> 00:26:31.480 align:middle line:84% So the light wave from, say, the lantern held by person B 00:26:31.480 --> 00:26:33.760 align:middle line:84% couldn't have either sped up or slowed down 00:26:33.760 --> 00:26:38.230 align:middle line:84% and had no choice but to travel along this worldline at exactly 00:26:38.230 --> 00:26:39.520 align:middle line:90% 45 degrees. 00:26:39.520 --> 00:26:42.910 align:middle line:84% Likewise for the light wave from my friend at position A. 00:26:42.910 --> 00:26:47.080 align:middle line:84% So if I know I'm equal distances along the x-axis from A and B 00:26:47.080 --> 00:26:50.350 align:middle line:84% And I receive both light waves at the same time here, 00:26:50.350 --> 00:26:53.180 align:middle line:84% then they must have been emitted simultaneously there. 00:26:53.180 --> 00:26:57.470 align:middle line:84% So now I can establish lines of simultaneity. 00:26:57.470 --> 00:27:00.610 align:middle line:84% Every point along this x-axis would 00:27:00.610 --> 00:27:05.000 align:middle line:84% have the same value of time in my coordinate system. 00:27:05.000 --> 00:27:08.490 align:middle line:84% The value of t on my t axis is the same. 00:27:08.490 --> 00:27:10.730 align:middle line:84% So the way I've set things up here, event 00:27:10.730 --> 00:27:13.280 align:middle line:84% A corresponds to the time t equals 00:27:13.280 --> 00:27:17.000 align:middle line:84% 0, event B corresponds to time equals 0 at time 00:27:17.000 --> 00:27:19.790 align:middle line:84% equals 0 sitting here on my chair. 00:27:19.790 --> 00:27:23.900 align:middle line:84% Likewise, every other moment line of simultaneity 00:27:23.900 --> 00:27:26.300 align:middle line:84% is parallel to the x-axis, they're 00:27:26.300 --> 00:27:29.000 align:middle line:84% all parallel to the line t equals 0. 00:27:29.000 --> 00:27:32.000 align:middle line:84% So I can imagine every event in my set of coordinates 00:27:32.000 --> 00:27:35.480 align:middle line:84% that had the assignment t equals 1 instead of t equals 0. 00:27:35.480 --> 00:27:38.210 align:middle line:84% All of these were simultaneous with each other, 00:27:38.210 --> 00:27:40.730 align:middle line:84% not simultaneous with events A and B, 00:27:40.730 --> 00:27:43.470 align:middle line:84% or every point along the line t equals 2 and so on. 00:27:43.470 --> 00:27:45.120 align:middle line:84% This is, I'm sure, very, very familiar. 00:27:45.120 --> 00:27:47.120 align:middle line:84% This is how we use graphs all the time. 00:27:47.120 --> 00:27:49.100 align:middle line:84% Minkowski's trying to just give us 00:27:49.100 --> 00:27:51.920 align:middle line:84% a more straightforward geometrical interpretation 00:27:51.920 --> 00:27:55.550 align:middle line:84% for what Einstein belabors in those opening pages about what 00:27:55.550 --> 00:27:57.260 align:middle line:84% do we mean by simultaneous events 00:27:57.260 --> 00:27:59.520 align:middle line:90% in our frame of reference. 00:27:59.520 --> 00:28:03.080 align:middle line:84% These are events that occur along a line of simultaneity. 00:28:03.080 --> 00:28:07.320 align:middle line:84% Well, now we can use our set of coordinates, our x and t, 00:28:07.320 --> 00:28:10.017 align:middle line:84% and map the motion of some other object that's 00:28:10.017 --> 00:28:11.100 align:middle line:90% moving with respect to us. 00:28:11.100 --> 00:28:14.220 align:middle line:84% For example, Einstein's favorite object, a train. 00:28:14.220 --> 00:28:17.130 align:middle line:84% So now we can chart the worldlines 00:28:17.130 --> 00:28:19.620 align:middle line:84% of the back of the train, the middle of the train, 00:28:19.620 --> 00:28:21.960 align:middle line:84% and the front of the train as the entire assembly 00:28:21.960 --> 00:28:26.400 align:middle line:84% moves past us while we stand still on the train platform. 00:28:26.400 --> 00:28:30.420 align:middle line:84% So this is our mapping of the moving object's motion 00:28:30.420 --> 00:28:33.000 align:middle line:84% from within our set of coordinates with respect-- 00:28:33.000 --> 00:28:35.240 align:middle line:84% the coordinates mark our frame of rest, 00:28:35.240 --> 00:28:36.990 align:middle line:84% and we're going to watch this moving train 00:28:36.990 --> 00:28:39.780 align:middle line:90% move through our coordinates. 00:28:39.780 --> 00:28:41.370 align:middle line:84% Now remember, we have a partner who 00:28:41.370 --> 00:28:42.450 align:middle line:84% rides in the middle of the train, 00:28:42.450 --> 00:28:44.610 align:middle line:84% again, very much like what Einstein was describing. 00:28:44.610 --> 00:28:46.110 align:middle line:84% So we have a colleague who's sitting 00:28:46.110 --> 00:28:47.235 align:middle line:90% in the middle of the train. 00:28:47.235 --> 00:28:49.560 align:middle line:84% She knows-- she's exactly equal distance from the front 00:28:49.560 --> 00:28:50.910 align:middle line:90% to the back of the train. 00:28:50.910 --> 00:28:53.610 align:middle line:84% She's asked friends on the train to shine their lanterns 00:28:53.610 --> 00:28:57.330 align:middle line:84% toward her and she'll be able to determine whether or not 00:28:57.330 --> 00:29:00.570 align:middle line:84% her colleagues A and B shine their lanterns toward her 00:29:00.570 --> 00:29:01.860 align:middle line:90% simultaneously. 00:29:01.860 --> 00:29:05.190 align:middle line:84% Well, light can only travel along 45 degrees 00:29:05.190 --> 00:29:07.290 align:middle line:84% on these diagrams for me or for her. 00:29:07.290 --> 00:29:10.230 align:middle line:84% Even on the moving train, the light wave 00:29:10.230 --> 00:29:13.020 align:middle line:84% that's emitted from point B doesn't speed up, 00:29:13.020 --> 00:29:16.830 align:middle line:84% it can only but go along a 45-degree angle. 00:29:16.830 --> 00:29:20.430 align:middle line:84% That's the real force of Einstein's second postulate. 00:29:20.430 --> 00:29:23.280 align:middle line:84% Likewise, the light wave that's emitted from point B 00:29:23.280 --> 00:29:25.540 align:middle line:84% can only travel at the speed of light. 00:29:25.540 --> 00:29:28.020 align:middle line:84% So in my coordinates, as much as in my colleagues 00:29:28.020 --> 00:29:30.210 align:middle line:84% coordinates on the train, those light waves 00:29:30.210 --> 00:29:35.140 align:middle line:84% must travel at 45-degree lines, at the constant speed of light. 00:29:35.140 --> 00:29:37.440 align:middle line:84% So our colleague who's moving on this train sitting 00:29:37.440 --> 00:29:40.545 align:middle line:84% in the middle, she receives the light beams from point A 00:29:40.545 --> 00:29:42.490 align:middle line:90% and point B at the same time. 00:29:42.490 --> 00:29:44.610 align:middle line:84% So now she knows those emission events 00:29:44.610 --> 00:29:46.072 align:middle line:90% must have been simultaneous. 00:29:46.072 --> 00:29:47.280 align:middle line:90% How could they not have been? 00:29:47.280 --> 00:29:49.350 align:middle line:84% She's an equal distance, the light waves 00:29:49.350 --> 00:29:51.790 align:middle line:84% couldn't have either sped up or slowed down. 00:29:51.790 --> 00:29:54.920 align:middle line:84% So therefore, all the points along this new line, 00:29:54.920 --> 00:30:00.090 align:middle line:84% we can call the x prime axis, all those events 00:30:00.090 --> 00:30:03.870 align:middle line:84% occur along a line of simultaneity for her. 00:30:03.870 --> 00:30:06.270 align:middle line:84% In the moving train, she knows exactly how 00:30:06.270 --> 00:30:08.490 align:middle line:84% to establish which events are simultaneous. 00:30:08.490 --> 00:30:10.440 align:middle line:84% She can take advantage of light waves, 00:30:10.440 --> 00:30:12.570 align:middle line:84% they travel at a universal speed. 00:30:12.570 --> 00:30:15.900 align:middle line:84% It just turns out that what events are simultaneous for her 00:30:15.900 --> 00:30:17.700 align:middle line:84% no longer match the set of events 00:30:17.700 --> 00:30:19.330 align:middle line:90% that are simultaneous for us. 00:30:19.330 --> 00:30:22.620 align:middle line:84% She has a different set of lines of simultaneity. 00:30:22.620 --> 00:30:24.420 align:middle line:84% And Minkowski was saying, in effect, 00:30:24.420 --> 00:30:27.750 align:middle line:84% you idiot, Einstein, it's just a coordinate rotation. 00:30:27.750 --> 00:30:30.060 align:middle line:84% You're just establishing a new set of coordinates 00:30:30.060 --> 00:30:32.920 align:middle line:84% like any student of geometry should be able to do. 00:30:32.920 --> 00:30:34.680 align:middle line:84% He said it only slightly more nicely 00:30:34.680 --> 00:30:37.160 align:middle line:90% in his published article. 00:30:37.160 --> 00:30:40.160 align:middle line:84% Meanwhile, the t prime axis is nothing 00:30:40.160 --> 00:30:44.000 align:middle line:84% other than the worldline of the zero point of space 00:30:44.000 --> 00:30:46.153 align:middle line:90% for the moving observer. 00:30:46.153 --> 00:30:47.570 align:middle line:84% That is to say, it's the worldline 00:30:47.570 --> 00:30:49.040 align:middle line:90% of the back of the train. 00:30:49.040 --> 00:30:51.470 align:middle line:84% The origin point of her coordinates 00:30:51.470 --> 00:30:54.740 align:middle line:84% is, say, the back of the train, x prime equals 0. 00:30:54.740 --> 00:30:57.470 align:middle line:84% She can measure any distance she wants with reference 00:30:57.470 --> 00:30:59.660 align:middle line:90% to the back of the train. 00:30:59.660 --> 00:31:01.580 align:middle line:90% She's moving with the train. 00:31:01.580 --> 00:31:03.470 align:middle line:84% She can always lay out meter sticks 00:31:03.470 --> 00:31:06.020 align:middle line:84% and measure how far the front of the train is from the back. 00:31:06.020 --> 00:31:08.810 align:middle line:84% So the origin, the spatial origin of her coordinates 00:31:08.810 --> 00:31:10.910 align:middle line:84% is the back of the train, how does that 00:31:10.910 --> 00:31:12.680 align:middle line:90% move through space and time? 00:31:12.680 --> 00:31:16.190 align:middle line:84% That just becomes her t prime axis. 00:31:16.190 --> 00:31:18.350 align:middle line:84% Much as our location of the point x 00:31:18.350 --> 00:31:21.410 align:middle line:84% equals 0 as it just sits still and moves 00:31:21.410 --> 00:31:25.340 align:middle line:84% through time, that maps out for us the t axis. 00:31:25.340 --> 00:31:29.090 align:middle line:84% The next thing Minkowski showed very easily for him 00:31:29.090 --> 00:31:32.090 align:middle line:84% as a geometer is that the angle between the x 00:31:32.090 --> 00:31:34.910 align:middle line:84% and the x prime axis, some angle theta, 00:31:34.910 --> 00:31:37.560 align:middle line:84% is exactly the same angle as that 00:31:37.560 --> 00:31:40.690 align:middle line:84% between the t and the t prime axis. 00:31:40.690 --> 00:31:44.400 align:middle line:84% In fact, that angle is directly related to this ratio v over c. 00:31:44.400 --> 00:31:46.560 align:middle line:84% So the faster the relative motion 00:31:46.560 --> 00:31:50.640 align:middle line:84% between us and the train, the more steeply these lines 00:31:50.640 --> 00:31:52.710 align:middle line:84% are inclined, the larger the angle. 00:31:52.710 --> 00:31:55.740 align:middle line:84% As the relative speed v rises with respect 00:31:55.740 --> 00:31:58.560 align:middle line:84% to that constant universal speed, the speed of light, 00:31:58.560 --> 00:32:01.320 align:middle line:84% this angle becomes larger, the x prime axis 00:32:01.320 --> 00:32:04.440 align:middle line:84% becomes tilted even more far away from x, 00:32:04.440 --> 00:32:08.070 align:middle line:84% the t prime axis gets tilted even more inward away from t. 00:32:08.070 --> 00:32:11.610 align:middle line:84% But all we're doing is establishing a new set of axes 00:32:11.610 --> 00:32:14.820 align:middle line:84% with respect to which we can mark the motion of objects 00:32:14.820 --> 00:32:17.340 align:middle line:84% through space and time according to the geometer 00:32:17.340 --> 00:32:19.380 align:middle line:90% Herman Minkowski. 00:32:19.380 --> 00:32:22.980 align:middle line:84% So now we come to some of these strange-sounding ideas 00:32:22.980 --> 00:32:26.070 align:middle line:84% that Einstein, again, had belabored as far as Minkowski 00:32:26.070 --> 00:32:26.940 align:middle line:90% was concerned. 00:32:26.940 --> 00:32:29.730 align:middle line:84% Einstein belabored with all this lengthy discussion 00:32:29.730 --> 00:32:34.890 align:middle line:84% of Ernst Mach-inspired measurement procedures. 00:32:34.890 --> 00:32:37.450 align:middle line:84% Minkowski says, it's just more geometry. 00:32:37.450 --> 00:32:39.690 align:middle line:84% Let's take the example of length contraction. 00:32:39.690 --> 00:32:43.110 align:middle line:84% We all agree, the procedure for measuring lengths 00:32:43.110 --> 00:32:45.540 align:middle line:84% is to measure the location of the front of the object 00:32:45.540 --> 00:32:47.950 align:middle line:84% and the back of the object at the same time 00:32:47.950 --> 00:32:49.260 align:middle line:90% and then take the difference. 00:32:49.260 --> 00:32:50.550 align:middle line:90% OK. 00:32:50.550 --> 00:32:52.980 align:middle line:84% So we measure the length of the train 00:32:52.980 --> 00:32:55.350 align:middle line:84% to be this length I've called L prime. 00:32:55.350 --> 00:32:58.740 align:middle line:84% We have our friend at the back of the train at position A, 00:32:58.740 --> 00:33:01.530 align:middle line:84% we have our friend at the front of the train at position B 00:33:01.530 --> 00:33:02.190 align:middle line:90% prime. 00:33:02.190 --> 00:33:05.040 align:middle line:84% We know those are simultaneous because they lie along 00:33:05.040 --> 00:33:08.640 align:middle line:84% a line of simultaneity for us, or a little bit more 00:33:08.640 --> 00:33:13.350 align:middle line:84% concretely, both the event A and the event B prime share 00:33:13.350 --> 00:33:15.000 align:middle line:90% the coordinate t equals 0. 00:33:15.000 --> 00:33:16.440 align:middle line:84% They have the same time coordinate 00:33:16.440 --> 00:33:18.760 align:middle line:84% in our coordinate system, in our reference frame. 00:33:18.760 --> 00:33:20.800 align:middle line:84% They lie along a line of simultaneity. 00:33:20.800 --> 00:33:22.990 align:middle line:84% So we've therefore marked the position 00:33:22.990 --> 00:33:25.510 align:middle line:84% along the x-axis of the front of the train 00:33:25.510 --> 00:33:28.570 align:middle line:84% and the back of the train at the same time in our reference 00:33:28.570 --> 00:33:29.180 align:middle line:90% frame. 00:33:29.180 --> 00:33:30.680 align:middle line:84% Now we can just take the difference, 00:33:30.680 --> 00:33:34.770 align:middle line:84% subtract the location of this point B prime along the x-axis, 00:33:34.770 --> 00:33:36.370 align:middle line:90% subtract it from the origin. 00:33:36.370 --> 00:33:37.060 align:middle line:90% OK. 00:33:37.060 --> 00:33:38.470 align:middle line:90% That's our length of the train. 00:33:38.470 --> 00:33:41.590 align:middle line:84% Front and back, same time, length L prime. 00:33:41.590 --> 00:33:44.830 align:middle line:84% Meanwhile, for our friend who's riding along the train, 00:33:44.830 --> 00:33:47.160 align:middle line:84% she knows exactly which events are simultaneous 00:33:47.160 --> 00:33:48.910 align:middle line:84% because she can have her friends test this 00:33:48.910 --> 00:33:50.560 align:middle line:84% with the exchange of light signals. 00:33:50.560 --> 00:33:54.430 align:middle line:84% And she knows very, very clearly that events A and B 00:33:54.430 --> 00:33:55.630 align:middle line:90% are simultaneous. 00:33:55.630 --> 00:33:58.480 align:middle line:84% For her reference frame, A and B lie along 00:33:58.480 --> 00:34:00.850 align:middle line:84% a shared line of simultaneity, just not 00:34:00.850 --> 00:34:02.470 align:middle line:90% our line of simultaneity. 00:34:02.470 --> 00:34:04.540 align:middle line:84% So her friend at the front of the train 00:34:04.540 --> 00:34:07.330 align:middle line:84% can mark that location, her friend at the back of the train 00:34:07.330 --> 00:34:08.409 align:middle line:90% can mark that location. 00:34:08.409 --> 00:34:11.530 align:middle line:84% We know they're doing it at the same time in the moving 00:34:11.530 --> 00:34:12.340 align:middle line:90% reference frame. 00:34:12.340 --> 00:34:14.949 align:middle line:84% Take the difference, it's the difference L, 00:34:14.949 --> 00:34:18.219 align:middle line:84% which, as you can see now, just trivially is longer than L 00:34:18.219 --> 00:34:19.040 align:middle line:90% prime. 00:34:19.040 --> 00:34:21.370 align:middle line:84% So what Einstein called-- and for that matter, 00:34:21.370 --> 00:34:23.679 align:middle line:84% Lorentz called length contraction, 00:34:23.679 --> 00:34:25.600 align:middle line:84% our measurement of the moving object 00:34:25.600 --> 00:34:27.850 align:middle line:84% is short compared to the measurement conducted 00:34:27.850 --> 00:34:29.679 align:middle line:84% by someone moving with that object. 00:34:29.679 --> 00:34:34.010 align:middle line:84% L prime is demonstrably shorter than L. And Minkowski says, 00:34:34.010 --> 00:34:35.830 align:middle line:90% in effect, of course it is. 00:34:35.830 --> 00:34:38.080 align:middle line:84% We're just projecting our measurements 00:34:38.080 --> 00:34:40.090 align:middle line:90% onto different sets of axes. 00:34:40.090 --> 00:34:42.580 align:middle line:84% We are geometers, we reckon objects 00:34:42.580 --> 00:34:44.060 align:middle line:84% with certain sets of coordinates, 00:34:44.060 --> 00:34:46.060 align:middle line:84% and we just happen to have different coordinates 00:34:46.060 --> 00:34:49.000 align:middle line:84% with which we're making sense of the world. 00:34:49.000 --> 00:34:51.940 align:middle line:84% He doesn't just derive the qualitative phenomenon 00:34:51.940 --> 00:34:54.850 align:middle line:84% of length contraction, he gets exactly the answer 00:34:54.850 --> 00:34:57.040 align:middle line:84% that both Einstein and Lorentz had gotten, 00:34:57.040 --> 00:35:00.220 align:middle line:84% even though they had very different forms of argument. 00:35:00.220 --> 00:35:02.320 align:middle line:84% He even gets the exact form of this gamma, 00:35:02.320 --> 00:35:05.650 align:middle line:84% because as you may remember, the angle here between x and x 00:35:05.650 --> 00:35:08.920 align:middle line:84% prime is directly related to v over c, 00:35:08.920 --> 00:35:11.860 align:middle line:84% as is the factor gamma, a few lines of algebra, 00:35:11.860 --> 00:35:14.830 align:middle line:84% not very hard, to see the relationship between this line 00:35:14.830 --> 00:35:16.840 align:middle line:90% segment and that line segment. 00:35:16.840 --> 00:35:19.780 align:middle line:84% What's important, again, to underscore is not the algebra-- 00:35:19.780 --> 00:35:21.610 align:middle line:84% I'm confident we can all do that. 00:35:21.610 --> 00:35:24.580 align:middle line:84% Is to say that for Minkowski, this is not about the ether, 00:35:24.580 --> 00:35:27.880 align:middle line:84% it's not about Machian observations that 00:35:27.880 --> 00:35:29.620 align:middle line:84% are subject to positive experience, 00:35:29.620 --> 00:35:32.218 align:middle line:84% this is just geometry as we transform 00:35:32.218 --> 00:35:33.760 align:middle line:84% between different coordinate systems. 00:35:33.760 --> 00:35:36.040 align:middle line:84% And you can almost hear him saying, "No duh!" 00:35:36.040 --> 00:35:39.310 align:middle line:84% to Einstein who cut all his classes. 00:35:39.310 --> 00:35:41.560 align:middle line:84% Minkowski goes on in this article, 00:35:41.560 --> 00:35:43.960 align:middle line:90% first worked out in 1907. 00:35:43.960 --> 00:35:46.120 align:middle line:84% Using these space-time diagrams, he then 00:35:46.120 --> 00:35:48.640 align:middle line:84% shows that the full Lorentz transformation 00:35:48.640 --> 00:35:50.040 align:middle line:90% was nothing but geometry. 00:35:50.040 --> 00:35:52.330 align:middle line:84% Remember, to Minkowski, everything was geometry. 00:35:52.330 --> 00:35:55.318 align:middle line:84% This was just a rotation in space-time. 00:35:55.318 --> 00:35:56.860 align:middle line:84% Now to warm up for that, again, we'll 00:35:56.860 --> 00:35:59.260 align:middle line:84% do something probably a little bit familiar to start. 00:35:59.260 --> 00:36:02.260 align:middle line:84% We'll talk about just an ordinary x-y plane. 00:36:02.260 --> 00:36:05.120 align:middle line:84% Two directions of space that are perpendicular to each other, 00:36:05.120 --> 00:36:08.200 align:middle line:84% the Cartesian plane, or, if you like the Euclidean plane. 00:36:08.200 --> 00:36:09.670 align:middle line:90% We can lay out our coordinates. 00:36:09.670 --> 00:36:13.180 align:middle line:84% x will be our horizontal axis, y is at right angles heading up 00:36:13.180 --> 00:36:14.030 align:middle line:90% the page. 00:36:14.030 --> 00:36:15.640 align:middle line:84% These are two directions of space. 00:36:15.640 --> 00:36:18.040 align:middle line:84% We can identify some points in the plane. 00:36:18.040 --> 00:36:20.410 align:middle line:84% We'll call that point P. We can label 00:36:20.410 --> 00:36:23.170 align:middle line:84% the location of that point in our coordinate system. 00:36:23.170 --> 00:36:25.000 align:middle line:90% We'll project what that means. 00:36:25.000 --> 00:36:28.300 align:middle line:84% Really, is we're projecting the location of point P 00:36:28.300 --> 00:36:31.100 align:middle line:90% onto our perpendicular axes. 00:36:31.100 --> 00:36:35.410 align:middle line:84% So we label the x-coordinate of point P as its projection along 00:36:35.410 --> 00:36:36.100 align:middle line:90% the P-- 00:36:36.100 --> 00:36:39.410 align:middle line:84% excuse me, along the x-axis, it has some value x1. 00:36:39.410 --> 00:36:42.170 align:middle line:84% We identify the location of point P, 00:36:42.170 --> 00:36:46.400 align:middle line:84% the projection along the y-axis, that's its y-coordinate y1. 00:36:46.400 --> 00:36:47.660 align:middle line:90% Fine. 00:36:47.660 --> 00:36:50.120 align:middle line:84% Now we all know-- even Einstein knew-- 00:36:50.120 --> 00:36:52.670 align:middle line:84% we're allowed to rotate our coordinate systems 00:36:52.670 --> 00:36:53.930 align:middle line:90% in the x-y plane. 00:36:53.930 --> 00:36:57.260 align:middle line:84% What if a friend of ours had used an inclined 00:36:57.260 --> 00:37:00.470 align:middle line:84% set of coordinates, x prime and y prime, 00:37:00.470 --> 00:37:02.870 align:middle line:84% that are rotated by some fixed angle 00:37:02.870 --> 00:37:05.150 align:middle line:84% away from our original coordinates? 00:37:05.150 --> 00:37:07.130 align:middle line:90% The point P hasn't moved. 00:37:07.130 --> 00:37:09.380 align:middle line:84% All we've done is change the labels 00:37:09.380 --> 00:37:11.300 align:middle line:90% we assigned to the coordinates. 00:37:11.300 --> 00:37:14.300 align:middle line:84% In our new coordinate system, in the rotated coordinates, 00:37:14.300 --> 00:37:17.030 align:middle line:84% the same point P, which itself hasn't moved, 00:37:17.030 --> 00:37:20.750 align:middle line:84% is assigned a different value of its x prime coordinate 00:37:20.750 --> 00:37:23.630 align:middle line:84% projecting now to the new axis and a different value 00:37:23.630 --> 00:37:27.260 align:middle line:84% of its y prime coordinate projecting to the new y-axis. 00:37:27.260 --> 00:37:29.900 align:middle line:84% These are related, quite straightforwardly, 00:37:29.900 --> 00:37:32.487 align:middle line:90% by a rotation matrix, by-- 00:37:32.487 --> 00:37:34.070 align:middle line:84% if we want to be a little fancy pants, 00:37:34.070 --> 00:37:37.400 align:middle line:84% we could say it's a one-parameter group 00:37:37.400 --> 00:37:40.100 align:middle line:84% of the rotation in two dimensions. 00:37:40.100 --> 00:37:42.200 align:middle line:84% But it's just a set of sines and cosines 00:37:42.200 --> 00:37:46.160 align:middle line:84% to take into account how the x and x prime axes are related 00:37:46.160 --> 00:37:48.620 align:middle line:84% to each other by this angle theta and the y 00:37:48.620 --> 00:37:49.850 align:middle line:90% and y prime axes. 00:37:49.850 --> 00:37:51.830 align:middle line:84% So it's actually very straightforward 00:37:51.830 --> 00:37:55.130 align:middle line:84% to relate the coordinate labels in one set of coordinates 00:37:55.130 --> 00:37:56.300 align:middle line:90% to those in the other. 00:37:56.300 --> 00:37:57.840 align:middle line:90% Nothing magic has happened. 00:37:57.840 --> 00:38:00.620 align:middle line:84% We can see that what I had called x prime 00:38:00.620 --> 00:38:03.890 align:middle line:84% was related to x in this very straightforward fashion, taking 00:38:03.890 --> 00:38:06.620 align:middle line:84% into account the different coordinate systems 00:38:06.620 --> 00:38:08.780 align:middle line:84% and their angle between them, and likewise, y 00:38:08.780 --> 00:38:11.420 align:middle line:90% prime in terms of x and y. 00:38:11.420 --> 00:38:13.070 align:middle line:84% So Minkowski says, that's all that's 00:38:13.070 --> 00:38:15.240 align:middle line:84% happening with space-time diagrams as well. 00:38:15.240 --> 00:38:18.110 align:middle line:84% Now let's imagine some event in space-time. 00:38:18.110 --> 00:38:20.690 align:middle line:84% So now we're going back to these Minkowski diagrams, 00:38:20.690 --> 00:38:23.720 align:middle line:84% the spatial direction runs along the horizontal axis, 00:38:23.720 --> 00:38:25.730 align:middle line:84% the time direction runs vertically, 00:38:25.730 --> 00:38:28.110 align:middle line:84% we have some event that happens in space and time. 00:38:28.110 --> 00:38:29.870 align:middle line:84% That is to say, we assign to it a time 00:38:29.870 --> 00:38:34.380 align:middle line:84% that happened at 12:00 noon at my house, a spatial location, 00:38:34.380 --> 00:38:35.810 align:middle line:90% some point. 00:38:35.810 --> 00:38:38.900 align:middle line:84% Now the observer on the train is going 00:38:38.900 --> 00:38:42.540 align:middle line:84% to label that exact same event, but give it different 00:38:42.540 --> 00:38:45.540 align:middle line:84% coordinate values because her coordinate system is different. 00:38:45.540 --> 00:38:48.750 align:middle line:84% The lines of simultaneity for her frame are different, 00:38:48.750 --> 00:38:51.630 align:middle line:84% the worldline of x equals 0 for her is different. 00:38:51.630 --> 00:38:54.150 align:middle line:84% So she has the x prime, t prime coordinate. 00:38:54.150 --> 00:38:57.960 align:middle line:84% She's still perfectly capable of labeling the event P 00:38:57.960 --> 00:39:00.960 align:middle line:84% and assigning to it values in her coordinates 00:39:00.960 --> 00:39:02.820 align:middle line:90% x1 prime and t1 prime. 00:39:02.820 --> 00:39:04.980 align:middle line:84% And again, remember, the angle of inclination 00:39:04.980 --> 00:39:07.297 align:middle line:84% here is directly related to the relative speed 00:39:07.297 --> 00:39:08.880 align:middle line:84% between these two sets of coordinates, 00:39:08.880 --> 00:39:11.160 align:middle line:90% these two reference frames. 00:39:11.160 --> 00:39:14.520 align:middle line:84% Moreover, what Lorenz had derived because of the-- 00:39:14.520 --> 00:39:17.160 align:middle line:84% what he assumed to be the elastic forces 00:39:17.160 --> 00:39:20.208 align:middle line:84% of an ether acting on the molecules, what Einstein had 00:39:20.208 --> 00:39:21.750 align:middle line:84% derived quite differently by thinking 00:39:21.750 --> 00:39:25.140 align:middle line:84% about the relativity of simultaneity, 00:39:25.140 --> 00:39:27.780 align:middle line:84% Minkowski says is just the set of rules 00:39:27.780 --> 00:39:31.650 align:middle line:84% for making rotations in this only slightly more complicated 00:39:31.650 --> 00:39:32.910 align:middle line:90% geometrical space. 00:39:32.910 --> 00:39:36.570 align:middle line:84% It's still basically like making rotations in the x-y plane. 00:39:36.570 --> 00:39:40.140 align:middle line:84% We have, again, to be fancy pants, a one-parameter family 00:39:40.140 --> 00:39:41.110 align:middle line:90% of rotations. 00:39:41.110 --> 00:39:44.650 align:middle line:84% The one parameter that changes now is the relative speed v, 00:39:44.650 --> 00:39:46.510 align:middle line:84% or if you like, the ratio of v over c, 00:39:46.510 --> 00:39:48.640 align:middle line:84% but of course, c is a universal constant. 00:39:48.640 --> 00:39:50.650 align:middle line:84% The only thing that really varies here 00:39:50.650 --> 00:39:52.810 align:middle line:90% is the speed little v. 00:39:52.810 --> 00:39:54.250 align:middle line:90% We have some other set of rules. 00:39:54.250 --> 00:39:56.890 align:middle line:84% These take the place of our sines and cosines 00:39:56.890 --> 00:39:58.640 align:middle line:90% of the relative angle. 00:39:58.640 --> 00:40:02.800 align:middle line:84% So once again, we can now trivially, geometrically 00:40:02.800 --> 00:40:05.830 align:middle line:84% relate the labels that the moving-- 00:40:05.830 --> 00:40:09.070 align:middle line:84% the person on the moving train had assigned to the same event 00:40:09.070 --> 00:40:12.610 align:middle line:84% P, relate those to the labels that we assigned standing still 00:40:12.610 --> 00:40:13.900 align:middle line:90% on the station platform. 00:40:13.900 --> 00:40:17.830 align:middle line:84% He gets precisely the Lorentz transformation, not because 00:40:17.830 --> 00:40:20.590 align:middle line:84% of the ether, not because of Einstein's 00:40:20.590 --> 00:40:23.360 align:middle line:84% arguments about how we perform measurements, 00:40:23.360 --> 00:40:26.470 align:middle line:84% but because he's just a geometer making rotations 00:40:26.470 --> 00:40:30.440 align:middle line:84% in a particular kind of two-dimensional space. 00:40:30.440 --> 00:40:32.500 align:middle line:84% So to Minkowski, the Lorentz transformation 00:40:32.500 --> 00:40:35.650 align:middle line:84% itself was nothing but geometric rotations in space-time. 00:40:35.650 --> 00:40:37.570 align:middle line:84% Again, so not surprising since we know 00:40:37.570 --> 00:40:40.480 align:middle line:84% Minkowski is a geometer above all. 00:40:40.480 --> 00:40:41.020 align:middle line:90% OK. 00:40:41.020 --> 00:40:43.330 align:middle line:84% Now this gives us something that's actually new. 00:40:43.330 --> 00:40:46.000 align:middle line:84% He's no longer just rederiving stuff in a new way. 00:40:46.000 --> 00:40:49.270 align:middle line:84% Now he gets stuff that neither Lorentz nor Einstein 00:40:49.270 --> 00:40:51.460 align:middle line:84% had actually recognized or realized. 00:40:51.460 --> 00:40:55.150 align:middle line:84% This next part comes really from Minkowski taking geometry 00:40:55.150 --> 00:40:56.710 align:middle line:90% very seriously. 00:40:56.710 --> 00:41:01.420 align:middle line:84% As he knew as a geometer, even in ordinary x-y plane-type 00:41:01.420 --> 00:41:05.080 align:middle line:84% geometry, when we can perform rotations, 00:41:05.080 --> 00:41:08.350 align:middle line:84% the coordinate labels of a given point will change, 00:41:08.350 --> 00:41:12.550 align:middle line:84% but some quantities remain unchanged even under rotations. 00:41:12.550 --> 00:41:14.570 align:middle line:90% For example, the distance. 00:41:14.570 --> 00:41:17.860 align:middle line:84% So what's the distance between the origin of my coordinates 00:41:17.860 --> 00:41:18.730 align:middle line:90% and the point P? 00:41:18.730 --> 00:41:20.290 align:middle line:90% We'll call that distance d. 00:41:20.290 --> 00:41:22.480 align:middle line:84% That doesn't change even if I change 00:41:22.480 --> 00:41:24.860 align:middle line:90% the labeling of my coordinates. 00:41:24.860 --> 00:41:27.730 align:middle line:84% So when I rotate by some angle theta, 00:41:27.730 --> 00:41:30.610 align:middle line:84% I change the labeling the description of x and y 00:41:30.610 --> 00:41:32.170 align:middle line:90% versus x prime and y prime. 00:41:32.170 --> 00:41:34.780 align:middle line:84% I didn't pick up the point P to move it further 00:41:34.780 --> 00:41:35.650 align:middle line:90% from the origin. 00:41:35.650 --> 00:41:38.350 align:middle line:84% The distance between, say, the origin and point P 00:41:38.350 --> 00:41:41.140 align:middle line:84% is invariant under those rotations. 00:41:41.140 --> 00:41:44.670 align:middle line:84% And again, it'll take you about 28 milliseconds 00:41:44.670 --> 00:41:46.920 align:middle line:84% to figure that out for the geometrical case just 00:41:46.920 --> 00:41:50.430 align:middle line:84% using sine squared theta plus cosine squared theta equals 1. 00:41:50.430 --> 00:41:52.770 align:middle line:84% Use the usual trigonometric identity 00:41:52.770 --> 00:41:55.290 align:middle line:84% between these trigonometric functions, 00:41:55.290 --> 00:41:56.760 align:middle line:84% and then it's very straightforward 00:41:56.760 --> 00:41:58.590 align:middle line:84% to see that the distance is invariant 00:41:58.590 --> 00:42:00.720 align:middle line:84% even though the coordinate labels have 00:42:00.720 --> 00:42:02.200 align:middle line:90% changed under rotation. 00:42:02.200 --> 00:42:05.430 align:middle line:84% That's what now becomes really new that Minkowski pushes 00:42:05.430 --> 00:42:09.060 align:middle line:84% forward as the first real concrete advantage 00:42:09.060 --> 00:42:12.330 align:middle line:84% as far as he's concerned of this geometrical approach. 00:42:12.330 --> 00:42:14.910 align:middle line:84% There's a generalization of distance 00:42:14.910 --> 00:42:16.680 align:middle line:90% that Minkowski introduces. 00:42:16.680 --> 00:42:19.230 align:middle line:84% It comes be called the space-time interval, 00:42:19.230 --> 00:42:21.180 align:middle line:84% often abbreviated by the letter S. 00:42:21.180 --> 00:42:24.480 align:middle line:84% It's like a version of distance for space-time 00:42:24.480 --> 00:42:29.620 align:middle line:84% instead of only a spatial problem, only space-space. 00:42:29.620 --> 00:42:33.595 align:middle line:84% So what is this space-time interval that remains unchanged 00:42:33.595 --> 00:42:35.220 align:middle line:84% even though the coordinate labels might 00:42:35.220 --> 00:42:37.440 align:middle line:84% be switched under these coordinate Lorentz 00:42:37.440 --> 00:42:38.590 align:middle line:90% transformations? 00:42:38.590 --> 00:42:40.680 align:middle line:84% So again, we come back to this way 00:42:40.680 --> 00:42:42.930 align:middle line:84% to relate our x and t coordinates 00:42:42.930 --> 00:42:45.400 align:middle line:84% to our x prime and t prime, that's what Minkowski 00:42:45.400 --> 00:42:47.300 align:middle line:90% is rederived from his rotation. 00:42:47.300 --> 00:42:50.530 align:middle line:84% And now he shows, using the definition of gamma, 00:42:50.530 --> 00:42:52.840 align:middle line:84% that just for the exact same reason 00:42:52.840 --> 00:42:55.690 align:middle line:84% that the Euclidean distance, little 00:42:55.690 --> 00:42:57.670 align:middle line:84% d, in the previous example remains 00:42:57.670 --> 00:43:01.090 align:middle line:84% unchanged, just because of the definitions of sine and cosine 00:43:01.090 --> 00:43:04.540 align:middle line:84% of a given angle, given the definition of gamma, that way 00:43:04.540 --> 00:43:06.800 align:middle line:84% of relating the kind of rotation, 00:43:06.800 --> 00:43:10.310 align:middle line:84% the degree to which x prime is inclined with respect to x, 00:43:10.310 --> 00:43:13.480 align:middle line:90% we get an unavoidable outcome. 00:43:13.480 --> 00:43:16.090 align:middle line:84% That whether we had chosen to label the event 00:43:16.090 --> 00:43:19.990 align:middle line:84% P in our x and t coordinates or x prime and t prime 00:43:19.990 --> 00:43:23.590 align:middle line:84% coordinates, the space-time interval, little s, 00:43:23.590 --> 00:43:25.750 align:middle line:90% has remained unchanged. 00:43:25.750 --> 00:43:28.420 align:middle line:84% Something remains invariant even though 00:43:28.420 --> 00:43:31.900 align:middle line:84% our different descriptions of spatial distances 00:43:31.900 --> 00:43:35.660 align:middle line:84% and temporal durations have changed. 00:43:35.660 --> 00:43:38.710 align:middle line:84% So to Minkowski, this shows that really, geometry-- 00:43:38.710 --> 00:43:42.320 align:middle line:84% in particular, the geometry of space-time, that, to Minkowski, 00:43:42.320 --> 00:43:44.400 align:middle line:84% becomes the only thing that matters. 00:43:44.400 --> 00:43:47.270 align:middle line:84% Not the ether, not this Mach-like 00:43:47.270 --> 00:43:50.360 align:middle line:84% show me how I can taste it or measure it or make 00:43:50.360 --> 00:43:52.430 align:middle line:84% it subject of positive experience, 00:43:52.430 --> 00:43:55.340 align:middle line:84% it's this geometrical feature of space-time. 00:43:55.340 --> 00:43:58.610 align:middle line:84% And in particular, not space and time separately. 00:43:58.610 --> 00:44:00.560 align:middle line:84% It's actually Minkowski's work, not 00:44:00.560 --> 00:44:02.870 align:middle line:84% Einstein's at first, that really introduces 00:44:02.870 --> 00:44:05.630 align:middle line:84% this notion of a union of space and time 00:44:05.630 --> 00:44:09.770 align:middle line:84% that very quickly becomes called by a single name, space-time. 00:44:09.770 --> 00:44:11.960 align:middle line:84% So that's what Minkowski is convinced 00:44:11.960 --> 00:44:15.380 align:middle line:84% is the only really important insight or outcome 00:44:15.380 --> 00:44:17.690 align:middle line:84% of Einstein's work, it is geometrical. 00:44:17.690 --> 00:44:20.070 align:middle line:84% So as he famously concludes in this article-- 00:44:20.070 --> 00:44:23.480 align:middle line:84% it was published soon after he passed away, 00:44:23.480 --> 00:44:25.770 align:middle line:84% laying out all this work on space-time diagrams, 00:44:25.770 --> 00:44:28.070 align:middle line:84% the invariant space-time interval and so on. 00:44:28.070 --> 00:44:32.690 align:middle line:84% He says, "Henceforth space by itself and time by itself 00:44:32.690 --> 00:44:35.240 align:middle line:84% are doomed to fade away into mere shadows." 00:44:35.240 --> 00:44:36.740 align:middle line:84% These are really just projections 00:44:36.740 --> 00:44:40.740 align:middle line:84% on a kind of idiosyncratic choice of coordinate axes. 00:44:40.740 --> 00:44:42.330 align:middle line:90% These are mere shadows. 00:44:42.330 --> 00:44:46.470 align:middle line:84% "And only a union of the two--" a single union of space-time-- 00:44:46.470 --> 00:44:48.450 align:middle line:90% "will preserve independence." 00:44:48.450 --> 00:44:51.460 align:middle line:84% So let me pause there again and ask for any questions. 00:44:51.460 --> 00:44:52.990 align:middle line:84% Any questions on the Minkowski stuff 00:44:52.990 --> 00:44:54.340 align:middle line:90% and on where he was coming from? 00:44:54.340 --> 00:44:59.420 align:middle line:90% 00:44:59.420 --> 00:45:02.178 align:middle line:84% So Silu asks, what were Einstein's reactions 00:45:02.178 --> 00:45:02.720 align:middle line:90% to Minkowski? 00:45:02.720 --> 00:45:03.690 align:middle line:90% Good, very good. 00:45:03.690 --> 00:45:06.320 align:middle line:84% So the first reaction of Einstein to Minkowski 00:45:06.320 --> 00:45:08.490 align:middle line:90% was also thoroughly predictable. 00:45:08.490 --> 00:45:09.420 align:middle line:90% He hated it. 00:45:09.420 --> 00:45:11.900 align:middle line:84% He never liked Minkowski as a person, he cut his classes. 00:45:11.900 --> 00:45:13.400 align:middle line:84% He was like, oh look, this guy still 00:45:13.400 --> 00:45:14.900 align:middle line:90% understands nothing important. 00:45:14.900 --> 00:45:19.160 align:middle line:90% The feeling was totally mutual. 00:45:19.160 --> 00:45:21.170 align:middle line:84% So Minkowski thought Einstein had 00:45:21.170 --> 00:45:23.983 align:middle line:84% made a mess of things and only grudgingly paid attention. 00:45:23.983 --> 00:45:25.400 align:middle line:84% Einstein read this paper and said, 00:45:25.400 --> 00:45:27.860 align:middle line:84% oh, this guy understands nothing, he's not reading Mach, 00:45:27.860 --> 00:45:29.870 align:middle line:84% he missed the real conceptual innovation. 00:45:29.870 --> 00:45:31.820 align:middle line:90% He's just playing with axes. 00:45:31.820 --> 00:45:34.820 align:middle line:84% Now the difference was, Minkowski died 00:45:34.820 --> 00:45:37.190 align:middle line:90% and Einstein didn't in 1908. 00:45:37.190 --> 00:45:39.840 align:middle line:84% So Einstein later, over-- and in fact, 00:45:39.840 --> 00:45:42.590 align:middle line:84% we'll look at this in our upcoming class session. 00:45:42.590 --> 00:45:44.870 align:middle line:90% Over the next several years-- 00:45:44.870 --> 00:45:49.310 align:middle line:84% again, about four years after Minkowski's work was published, 00:45:49.310 --> 00:45:53.240 align:middle line:84% did Einstein slowly, grudgingly, with other kind of inputs 00:45:53.240 --> 00:45:56.390 align:middle line:84% and nudges, begin to appreciate the geometrical view 00:45:56.390 --> 00:45:58.620 align:middle line:84% that Minkowski was putting forward. 00:45:58.620 --> 00:46:00.150 align:middle line:84% Einstein's original reaction was, 00:46:00.150 --> 00:46:01.503 align:middle line:90% this guy still doesn't get it. 00:46:01.503 --> 00:46:03.670 align:middle line:84% I would cut his classes again if I were still there. 00:46:03.670 --> 00:46:05.940 align:middle line:84% I mean, really, it was just this mutual, like, blah. 00:46:05.940 --> 00:46:07.980 align:middle line:84% And it's really about four years later, 00:46:07.980 --> 00:46:11.790 align:middle line:84% as we'll talk a bit about in the next class, that Einstein 00:46:11.790 --> 00:46:15.420 align:middle line:84% grudgingly comes along to say, geometry is really cool, 00:46:15.420 --> 00:46:18.820 align:middle line:84% space-time as a unified notion actually seems important. 00:46:18.820 --> 00:46:21.240 align:middle line:84% And then he says, oh, I wish I had cited all that math 00:46:21.240 --> 00:46:24.090 align:middle line:84% after all, and he goes back to one of his friends, Marcel 00:46:24.090 --> 00:46:26.507 align:middle line:84% Grossmann, and says, oh, not only can I borrow your notes, 00:46:26.507 --> 00:46:28.340 align:middle line:84% can you teach me all the stuff I should have 00:46:28.340 --> 00:46:29.610 align:middle line:90% learned when I was in college? 00:46:29.610 --> 00:46:32.580 align:middle line:84% Lesson number 3 is don't act like Einstein. 00:46:32.580 --> 00:46:34.263 align:middle line:84% And Gary asks, if Minkowski had lived, 00:46:34.263 --> 00:46:36.180 align:middle line:84% who would have been deemed the greater genius? 00:46:36.180 --> 00:46:38.140 align:middle line:84% Oh, that's an interesting question. 00:46:38.140 --> 00:46:39.900 align:middle line:90% I don't know. 00:46:39.900 --> 00:46:41.910 align:middle line:90% That's a time-varying question. 00:46:41.910 --> 00:46:44.430 align:middle line:90% In 1909, still Minkowski. 00:46:44.430 --> 00:46:46.800 align:middle line:90% In 1915, I'm not sure. 00:46:46.800 --> 00:46:49.100 align:middle line:90% In 1945, I don't know. 00:46:49.100 --> 00:46:50.100 align:middle line:90% It's interesting to see. 00:46:50.100 --> 00:46:52.800 align:middle line:84% And I imagine the mathematicians then, 00:46:52.800 --> 00:46:56.190 align:middle line:84% as now, would herald Minkowski's work for a lot longer. 00:46:56.190 --> 00:46:58.890 align:middle line:84% We'd see, I'm sure, some different opinions continue 00:46:58.890 --> 00:47:00.600 align:middle line:90% to vary. 00:47:00.600 --> 00:47:02.112 align:middle line:84% Jesus writes, it's scary to realize 00:47:02.112 --> 00:47:04.320 align:middle line:84% that nobody cared too much about these groundbreaking 00:47:04.320 --> 00:47:05.380 align:middle line:90% results for so long. 00:47:05.380 --> 00:47:07.380 align:middle line:84% Makes me wonder what kind of groundbreaking work 00:47:07.380 --> 00:47:08.880 align:middle line:84% may have actually fallen through the cracks 00:47:08.880 --> 00:47:10.672 align:middle line:84% and never gotten a full exploration at all. 00:47:10.672 --> 00:47:13.830 align:middle line:90% Yes, that's a fair point. 00:47:13.830 --> 00:47:15.330 align:middle line:84% That's something we can keep in mind 00:47:15.330 --> 00:47:17.160 align:middle line:90% throughout this whole semester. 00:47:17.160 --> 00:47:21.810 align:middle line:84% Not only-- there's all kinds of gatekeeping 00:47:21.810 --> 00:47:25.260 align:middle line:84% about who determines at what time what work is worth 00:47:25.260 --> 00:47:26.400 align:middle line:90% paying attention to. 00:47:26.400 --> 00:47:29.920 align:middle line:84% And institutions play a role, individual rivalries 00:47:29.920 --> 00:47:30.780 align:middle line:90% play a role. 00:47:30.780 --> 00:47:33.300 align:middle line:84% Soon, we'll come to examples where worldly events play 00:47:33.300 --> 00:47:35.760 align:middle line:84% a role, well beyond the control of any given 00:47:35.760 --> 00:47:37.800 align:middle line:90% individual or institution. 00:47:37.800 --> 00:47:40.380 align:middle line:84% Sexism plays a role, racism plays a role. 00:47:40.380 --> 00:47:43.380 align:middle line:84% I mean, there's all kinds of things that factor into who 00:47:43.380 --> 00:47:45.030 align:middle line:84% determines this is of value and worth 00:47:45.030 --> 00:47:48.010 align:middle line:84% paying attention to at any given time. 00:47:48.010 --> 00:47:49.950 align:middle line:84% And that's a general historical lesson 00:47:49.950 --> 00:47:51.845 align:middle line:84% we grapple with all the time, not just 00:47:51.845 --> 00:47:53.970 align:middle line:84% in the history of science, but much more generally. 00:47:53.970 --> 00:47:55.890 align:middle line:84% And we'll have a lot of examples of that. 00:47:55.890 --> 00:47:57.600 align:middle line:84% Really, actually, more examples even 00:47:57.600 --> 00:48:00.320 align:middle line:84% pretty soon in our own class, alone examples 00:48:00.320 --> 00:48:03.440 align:middle line:84% we might think of from other instances. 00:48:03.440 --> 00:48:06.290 align:middle line:84% Here's an example where, to his credit, 00:48:06.290 --> 00:48:08.570 align:middle line:84% Einstein later, grudgingly, came around 00:48:08.570 --> 00:48:10.370 align:middle line:84% to say, oh, that stuff is important, 00:48:10.370 --> 00:48:12.110 align:middle line:90% Minkowski does deserve credit. 00:48:12.110 --> 00:48:13.640 align:middle line:84% I now see it was really valuable. 00:48:13.640 --> 00:48:16.550 align:middle line:84% But it took Einstein himself quite a few years to get there. 00:48:16.550 --> 00:48:20.810 align:middle line:84% Doesn't always happen, or there can be a much longer delay. 00:48:20.810 --> 00:48:25.400 align:middle line:84% Any other questions on Minkowski's reaction to 00:48:25.400 --> 00:48:27.890 align:middle line:90% or remaking of Einstein's work? 00:48:27.890 --> 00:48:30.690 align:middle line:90% 00:48:30.690 --> 00:48:33.563 align:middle line:84% So Minkowski actually thought this was really a big deal. 00:48:33.563 --> 00:48:35.730 align:middle line:84% Minkowski thought his own work was a big deal, which 00:48:35.730 --> 00:48:36.622 align:middle line:90% is not shocking. 00:48:36.622 --> 00:48:38.580 align:middle line:84% It's also not unique in the history of science. 00:48:38.580 --> 00:48:40.580 align:middle line:84% And in particular, he thought it wasn't actually 00:48:40.580 --> 00:48:43.500 align:middle line:84% only about coordinate transformations and rotations 00:48:43.500 --> 00:48:44.040 align:middle line:90% of axes. 00:48:44.040 --> 00:48:47.290 align:middle line:84% He thought that was the most obvious thing to do, 00:48:47.290 --> 00:48:49.230 align:middle line:84% and anyone who doesn't do that is a dummy. 00:48:49.230 --> 00:48:52.320 align:middle line:84% He had very clear ideas about the proper tools of research. 00:48:52.320 --> 00:48:54.270 align:middle line:84% But even in that first article that 00:48:54.270 --> 00:48:56.370 align:middle line:84% gets published soon after he passed away, 00:48:56.370 --> 00:49:00.720 align:middle line:84% he develops in the later sections a larger worldview. 00:49:00.720 --> 00:49:02.820 align:middle line:84% Literally, a view of the world that's 00:49:02.820 --> 00:49:05.880 align:middle line:84% based on this union space-time notion. 00:49:05.880 --> 00:49:09.900 align:middle line:84% It gets very wrapped up with certain trends in idealist 00:49:09.900 --> 00:49:10.950 align:middle line:90% German philosophy. 00:49:10.950 --> 00:49:14.550 align:middle line:84% He's very inspired by the work of Immanuel Kant, for example. 00:49:14.550 --> 00:49:16.510 align:middle line:84% Einstein was, too, but in different ways. 00:49:16.510 --> 00:49:18.660 align:middle line:84% And so as far as Minkowski is concerned, 00:49:18.660 --> 00:49:21.750 align:middle line:84% he thought he'd found a new deep level of reality. 00:49:21.750 --> 00:49:25.380 align:middle line:84% The reality wasn't the ether to Minkowski. 00:49:25.380 --> 00:49:28.740 align:middle line:84% The reality was there was this single thing, space-time, 00:49:28.740 --> 00:49:31.630 align:middle line:84% and there are certain things on which all people will agree, 00:49:31.630 --> 00:49:34.480 align:middle line:84% these invariances, even as we disagree separately 00:49:34.480 --> 00:49:38.060 align:middle line:84% on measurements of length or measurements of time. 00:49:38.060 --> 00:49:40.690 align:middle line:84% And so that, for him, reintroduced a kind 00:49:40.690 --> 00:49:43.090 align:middle line:90% of absolute, an absolutism. 00:49:43.090 --> 00:49:46.870 align:middle line:84% Not absolute rest like in Newton, not absolute space, 00:49:46.870 --> 00:49:49.630 align:middle line:84% because space and time, he says, are now just mere shadows. 00:49:49.630 --> 00:49:53.180 align:middle line:84% But there is an absolute past and an absolute future. 00:49:53.180 --> 00:49:55.930 align:middle line:84% There's an absolute away, the things that we would now call, 00:49:55.930 --> 00:49:58.330 align:middle line:84% if you've had some more coursework in this area, 00:49:58.330 --> 00:50:01.870 align:middle line:84% you'd recognize this things like spacelike-separated and so on. 00:50:01.870 --> 00:50:05.590 align:middle line:84% He starts mapping sets of events that could not possibly 00:50:05.590 --> 00:50:07.750 align:middle line:84% have exchanged even a single light signal yet. 00:50:07.750 --> 00:50:10.000 align:middle line:84% Those are absolutely separated from each other. 00:50:10.000 --> 00:50:13.150 align:middle line:84% And he gives these very absolutist language to them. 00:50:13.150 --> 00:50:15.130 align:middle line:84% So he's actually mapping what he considers 00:50:15.130 --> 00:50:18.520 align:middle line:84% like the ultimate absolute structure of space-time. 00:50:18.520 --> 00:50:20.710 align:middle line:84% And as far as he's concerned, that's the real lesson 00:50:20.710 --> 00:50:22.660 align:middle line:90% that the geometry leads us to. 00:50:22.660 --> 00:50:25.810 align:middle line:84% So he actually thinks it's not a theory of relativity at all. 00:50:25.810 --> 00:50:28.360 align:middle line:84% Relativity is like an accident and boring 00:50:28.360 --> 00:50:30.250 align:middle line:84% and trivial and the stupid thing that no one 00:50:30.250 --> 00:50:31.630 align:middle line:90% should pay attention to. 00:50:31.630 --> 00:50:33.940 align:middle line:84% Relativity-- oh, we differ, the person on the train 00:50:33.940 --> 00:50:35.060 align:middle line:90% says this wasn't the same. 00:50:35.060 --> 00:50:37.120 align:middle line:90% He says, that's just accidents. 00:50:37.120 --> 00:50:38.080 align:middle line:90% Who cares? 00:50:38.080 --> 00:50:40.960 align:middle line:84% What's actually important are these underlying invariances 00:50:40.960 --> 00:50:41.900 align:middle line:90% or absolutes. 00:50:41.900 --> 00:50:44.770 align:middle line:84% We all agree on certain combinations of things, 00:50:44.770 --> 00:50:46.460 align:middle line:90% and that's the real lesson. 00:50:46.460 --> 00:50:51.160 align:middle line:84% So for him, this becomes a re-injection of the absolute. 00:50:51.160 --> 00:50:53.380 align:middle line:84% Not because for all sitting in a physical ether. 00:50:53.380 --> 00:50:55.210 align:middle line:84% He couldn't care less, he's a geometer. 00:50:55.210 --> 00:50:58.360 align:middle line:84% Because there is a unique structure 00:50:58.360 --> 00:51:01.090 align:middle line:84% to space-time that the tools of geometry 00:51:01.090 --> 00:51:05.050 align:middle line:84% are uniquely well-adapted to help us understand. 00:51:05.050 --> 00:51:06.657 align:middle line:90% So does that makes sense? 00:51:06.657 --> 00:51:07.990 align:middle line:90% Does that address your question? 00:51:07.990 --> 00:51:08.600 align:middle line:90% OK, cool. 00:51:08.600 --> 00:51:09.120 align:middle line:90% Thank you. 00:51:09.120 --> 00:51:09.662 align:middle line:90% So let's see. 00:51:09.662 --> 00:51:11.450 align:middle line:84% Fisher asks the question in the chat. 00:51:11.450 --> 00:51:13.700 align:middle line:84% We now think of space-time as something is deformable. 00:51:13.700 --> 00:51:14.900 align:middle line:90% Hey, that's cheating. 00:51:14.900 --> 00:51:16.070 align:middle line:84% That's-- we're going to get there next week. 00:51:16.070 --> 00:51:17.070 align:middle line:90% You don't know that yet. 00:51:17.070 --> 00:51:18.680 align:middle line:90% It's 1905 or 1908. 00:51:18.680 --> 00:51:21.410 align:middle line:84% But certainly, we'll talk about quite a lot starting 00:51:21.410 --> 00:51:24.830 align:middle line:84% in this coming-- the next class session where Einstein himself 00:51:24.830 --> 00:51:27.110 align:middle line:84% gets to is to begin thinking about space-time as one 00:51:27.110 --> 00:51:31.220 align:middle line:84% thing that also can respond, to be deformable. 00:51:31.220 --> 00:51:33.333 align:middle line:84% Originally introduced as a mathematical artifact, 00:51:33.333 --> 00:51:35.000 align:middle line:84% was a point where we moved from thinking 00:51:35.000 --> 00:51:36.828 align:middle line:84% about it as a mathematical concept when 00:51:36.828 --> 00:51:38.120 align:middle line:90% general relavity was published? 00:51:38.120 --> 00:51:38.745 align:middle line:90% Yes, very good. 00:51:38.745 --> 00:51:40.938 align:middle line:84% OK, so Fisher, that's a great question. 00:51:40.938 --> 00:51:41.730 align:middle line:90% Thank you for that. 00:51:41.730 --> 00:51:43.190 align:middle line:84% And we're going to come-- we'll get to sit with that 00:51:43.190 --> 00:51:45.773 align:middle line:84% actually quite directly on our next class, so I'm going to I'm 00:51:45.773 --> 00:51:47.480 align:middle line:90% going to punt on that for now. 00:51:47.480 --> 00:51:50.450 align:middle line:84% The short answer is it's not-- it doesn't come only 00:51:50.450 --> 00:51:52.370 align:middle line:84% when Einstein publishes general relativity. 00:51:52.370 --> 00:51:55.160 align:middle line:90% That comes very late, 1915. 00:51:55.160 --> 00:51:58.730 align:middle line:84% But it's an evolution in Einstein's own thinking 00:51:58.730 --> 00:52:00.700 align:middle line:84% with some colleagues, like Marcel Grossman 00:52:00.700 --> 00:52:04.080 align:middle line:84% and a small circle, over the intervening decade. 00:52:04.080 --> 00:52:07.770 align:middle line:84% So starting around 1911, '12, '13, 00:52:07.770 --> 00:52:10.110 align:middle line:84% he has a lot of mistakes, a lot of blind alleys. 00:52:10.110 --> 00:52:12.690 align:middle line:84% He starts wondering about that more in the terms 00:52:12.690 --> 00:52:14.580 align:middle line:90% as you describe it. 00:52:14.580 --> 00:52:19.260 align:middle line:84% Starting well downstream both from his own work in 1905, 00:52:19.260 --> 00:52:21.930 align:middle line:84% downstream by four years even from Minkowski's work. 00:52:21.930 --> 00:52:24.150 align:middle line:84% It's a real intellectual journey. 00:52:24.150 --> 00:52:26.550 align:middle line:84% So that's a preview, we'll get there next time. 00:52:26.550 --> 00:52:27.660 align:middle line:90% Great. 00:52:27.660 --> 00:52:30.030 align:middle line:84% Gary asks, how did folks react to Minkowski's work? 00:52:30.030 --> 00:52:31.380 align:middle line:90% Again, I think maybe-- 00:52:31.380 --> 00:52:32.940 align:middle line:84% sorry, maybe I mentioned that or you 00:52:32.940 --> 00:52:34.260 align:middle line:90% asked earlier, who was the-- 00:52:34.260 --> 00:52:35.910 align:middle line:84% who made the greater contribution? 00:52:35.910 --> 00:52:38.010 align:middle line:84% Yeah, Minkowski was on people's radar screen. 00:52:38.010 --> 00:52:40.170 align:middle line:84% He was a big fancy senior mathematician, 00:52:40.170 --> 00:52:43.530 align:middle line:84% very widely published, very influential at a central-- 00:52:43.530 --> 00:52:45.400 align:middle line:84% center of learning in Western Europe. 00:52:45.400 --> 00:52:47.640 align:middle line:84% So his work was paid attention to mostly 00:52:47.640 --> 00:52:52.110 align:middle line:84% by other mathematicians and by some mathematical physicists 00:52:52.110 --> 00:52:55.035 align:middle line:90% like Hendrik Lorentz. 00:52:55.035 --> 00:52:56.910 align:middle line:84% Not every single physicist, and certainly not 00:52:56.910 --> 00:52:59.730 align:middle line:84% every experimental physicist, but it was not 00:52:59.730 --> 00:53:01.810 align:middle line:84% buried and forgotten for years and years. 00:53:01.810 --> 00:53:04.140 align:middle line:84% It was appreciated in certain specialist communities 00:53:04.140 --> 00:53:07.020 align:middle line:84% right away, and then as we'll talk about later, 00:53:07.020 --> 00:53:10.230 align:middle line:84% even other people like Einstein belatedly 00:53:10.230 --> 00:53:11.320 align:middle line:90% come to admire it as well. 00:53:11.320 --> 00:53:14.010 align:middle line:84% So it becomes very, very well-known, partly, I think, 00:53:14.010 --> 00:53:15.910 align:middle line:84% because Einstein goes back to it. 00:53:15.910 --> 00:53:17.413 align:middle line:84% And Einstein himself becomes, then-- 00:53:17.413 --> 00:53:20.080 align:middle line:84% eventually someone that everyone pays attention to, many people. 00:53:20.080 --> 00:53:22.620 align:middle line:90% So it has a bumpy few years. 00:53:22.620 --> 00:53:24.817 align:middle line:84% But it was not launched into obscurity 00:53:24.817 --> 00:53:26.400 align:middle line:84% because Minkowski himself was actually 00:53:26.400 --> 00:53:30.000 align:middle line:84% already quite well-established and well-known. 00:53:30.000 --> 00:53:33.160 align:middle line:84% Abdul Aziz asks, what was Lorenz's stance with regard 00:53:33.160 --> 00:53:33.660 align:middle line:90% to this? 00:53:33.660 --> 00:53:33.930 align:middle line:90% Good. 00:53:33.930 --> 00:53:35.430 align:middle line:84% Did he jump on the space-time wagon? 00:53:35.430 --> 00:53:35.930 align:middle line:90% Great. 00:53:35.930 --> 00:53:37.380 align:middle line:84% So the short answer is-- we'll see 00:53:37.380 --> 00:53:38.850 align:middle line:84% an example of the kind of reaction 00:53:38.850 --> 00:53:40.860 align:middle line:84% actually the next part of today's class. 00:53:40.860 --> 00:53:43.470 align:middle line:84% Not Lorentz's, per se, but an example. 00:53:43.470 --> 00:53:46.770 align:middle line:84% Lorentz, really, for the rest of his career 00:53:46.770 --> 00:53:49.650 align:middle line:84% was still pretty convinced that the ether exists. 00:53:49.650 --> 00:53:52.500 align:middle line:84% And he had a great appreciation for some of these very clever 00:53:52.500 --> 00:53:54.000 align:middle line:90% mathematical techniques. 00:53:54.000 --> 00:53:56.310 align:middle line:84% He certainly liked it that Einstein's, and now 00:53:56.310 --> 00:53:59.040 align:middle line:84% Minkowski's work, led back to equations that he 00:53:59.040 --> 00:54:02.700 align:middle line:84% himself had derived from his own set of arguments. 00:54:02.700 --> 00:54:04.380 align:middle line:84% But he really was convinced that there's 00:54:04.380 --> 00:54:06.750 align:middle line:84% something physical about this elastic medium in which 00:54:06.750 --> 00:54:08.860 align:middle line:84% for all immersed for a long, long time. 00:54:08.860 --> 00:54:12.900 align:middle line:84% So he thought these were more cool tools, 00:54:12.900 --> 00:54:15.055 align:middle line:84% more mathematical tools with which 00:54:15.055 --> 00:54:17.430 align:middle line:84% to keep asking questions he already considered important. 00:54:17.430 --> 00:54:19.378 align:middle line:84% Those weren't, as we've seen, the same set 00:54:19.378 --> 00:54:21.420 align:middle line:84% of questions that Einstein thought were important 00:54:21.420 --> 00:54:23.020 align:middle line:84% or that Minkowski thought were important. 00:54:23.020 --> 00:54:25.187 align:middle line:84% And in that sense, it's a lot like the example we'll 00:54:25.187 --> 00:54:26.820 align:middle line:90% look at now from Cambridge. 00:54:26.820 --> 00:54:30.690 align:middle line:90% 00:54:30.690 --> 00:54:31.702 align:middle line:90% Yeah, very good. 00:54:31.702 --> 00:54:33.660 align:middle line:84% So again, we'll look at this much more directly 00:54:33.660 --> 00:54:36.540 align:middle line:84% in the upcoming class session next week. 00:54:36.540 --> 00:54:39.270 align:middle line:84% The short answer is, it wasn't really Minkowski, per se, 00:54:39.270 --> 00:54:41.450 align:middle line:84% because he just had no respect for Minkowski, 00:54:41.450 --> 00:54:42.990 align:middle line:90% he kept cutting his classes. 00:54:42.990 --> 00:54:47.040 align:middle line:84% It was actually an independent series of thought experiments 00:54:47.040 --> 00:54:49.920 align:middle line:84% and conceptual puzzles and calculations 00:54:49.920 --> 00:54:52.170 align:middle line:84% that Einstein got immersed in, and he 00:54:52.170 --> 00:54:56.280 align:middle line:84% kept getting stuck, and that led him back to realizing geometry 00:54:56.280 --> 00:54:58.440 align:middle line:84% might be really important, by which point, 00:54:58.440 --> 00:55:01.530 align:middle line:84% Minkowski had died, he went to his former classmate, 00:55:01.530 --> 00:55:04.380 align:middle line:84% his buddy, Marcel Grossman, and had 00:55:04.380 --> 00:55:06.660 align:middle line:84% to do a crash course on the stuff 00:55:06.660 --> 00:55:08.310 align:middle line:84% that he kind of should have learned 00:55:08.310 --> 00:55:09.760 align:middle line:90% when he was an undergraduate. 00:55:09.760 --> 00:55:12.120 align:middle line:84% So it was-- he came to an appreciation 00:55:12.120 --> 00:55:14.610 align:middle line:84% of the geometrical approach, of Minkowski's work 00:55:14.610 --> 00:55:16.860 align:middle line:84% in particular thinking about a single object called 00:55:16.860 --> 00:55:17.520 align:middle line:90% space-time. 00:55:17.520 --> 00:55:18.323 align:middle line:90% He gets there. 00:55:18.323 --> 00:55:20.490 align:middle line:84% But not because he reads Minkowski's paper and says, 00:55:20.490 --> 00:55:22.620 align:middle line:84% oh, my great professor, I'm totally convinced. 00:55:22.620 --> 00:55:25.110 align:middle line:84% He's like, that's dumb, it's wrong, it's boring, 00:55:25.110 --> 00:55:27.660 align:middle line:84% he's still irrelevant, I would cut his classes again. 00:55:27.660 --> 00:55:30.000 align:middle line:84% That's the reaction for like four years, three 00:55:30.000 --> 00:55:31.040 align:middle line:90% or four years. 00:55:31.040 --> 00:55:36.010 align:middle line:84% And it's a separate set of lines of thought that-- 00:55:36.010 --> 00:55:37.580 align:middle line:84% and interactions with other people, 00:55:37.580 --> 00:55:40.210 align:middle line:84% including friends who were better-versed in mathematics, 00:55:40.210 --> 00:55:42.820 align:middle line:84% that bring him his own thinking back-- 00:55:42.820 --> 00:55:46.280 align:middle line:84% or bring his thinking towards a more thoroughly geometric 00:55:46.280 --> 00:55:46.780 align:middle line:90% approach. 00:55:46.780 --> 00:55:47.830 align:middle line:90% And then he never leaves. 00:55:47.830 --> 00:55:51.220 align:middle line:84% Then he becomes a card-carrying geometer himself 00:55:51.220 --> 00:55:54.730 align:middle line:84% once he gets a crash course from real geometers 00:55:54.730 --> 00:55:58.330 align:middle line:84% and gets a newfound appreciation because of his own questions 00:55:58.330 --> 00:55:59.200 align:middle line:90% have led him there. 00:55:59.200 --> 00:56:02.740 align:middle line:84% So he comes to greatly admire Minkowski's work, 00:56:02.740 --> 00:56:05.710 align:middle line:84% and indeed, to try to build upon it, as we'll see next time, 00:56:05.710 --> 00:56:08.380 align:middle line:84% but not because he thought, oh, this is 00:56:08.380 --> 00:56:09.910 align:middle line:90% a cool work by a smart person. 00:56:09.910 --> 00:56:13.430 align:middle line:84% It was, again, a kind of roundabout thing. 00:56:13.430 --> 00:56:15.760 align:middle line:84% And so-- and we'll look at that in a bit more detail 00:56:15.760 --> 00:56:16.765 align:middle line:90% in the coming class. 00:56:16.765 --> 00:56:18.140 align:middle line:84% Any other question on this stuff? 00:56:18.140 --> 00:56:20.473 align:middle line:84% If not, I'll go to this last part of the lecture on some 00:56:20.473 --> 00:56:23.140 align:middle line:84% of the pretty wacky work that the Cambridge gang 00:56:23.140 --> 00:56:24.640 align:middle line:90% starts doing with this as well. 00:56:24.640 --> 00:56:30.010 align:middle line:90% 00:56:30.010 --> 00:56:31.030 align:middle line:90% Yeah, thank you, Fisher. 00:56:31.030 --> 00:56:33.430 align:middle line:84% So as far as Minkowski was concerned, 00:56:33.430 --> 00:56:35.643 align:middle line:84% Einstein's paper, especially the first few pages, 00:56:35.643 --> 00:56:37.810 align:middle line:84% which you are all dutifully reviewing and maybe even 00:56:37.810 --> 00:56:41.080 align:middle line:84% rejecting, or not, or asking at least for revisions, 00:56:41.080 --> 00:56:43.960 align:middle line:84% the first few pages, at least, are really, 00:56:43.960 --> 00:56:47.630 align:middle line:84% to someone like Minkowski, a jumble word salad. 00:56:47.630 --> 00:56:53.200 align:middle line:84% This is like, of course I know how to calculate distance 00:56:53.200 --> 00:56:55.600 align:middle line:84% traveled if the person's traveling at constant speed. 00:56:55.600 --> 00:56:57.260 align:middle line:90% That's like middle school. 00:56:57.260 --> 00:56:59.680 align:middle line:84% This is not like, why are you belaboring this paragraph 00:56:59.680 --> 00:57:00.700 align:middle line:90% after paragraph? 00:57:00.700 --> 00:57:03.940 align:middle line:84% Why are you using this Mach-inspired language 00:57:03.940 --> 00:57:06.310 align:middle line:84% without even citing Ernst Mach about how 00:57:06.310 --> 00:57:09.627 align:middle line:84% objects should be subject to empirical measurement? 00:57:09.627 --> 00:57:11.710 align:middle line:84% Why are you telling me how to have my friend shine 00:57:11.710 --> 00:57:12.640 align:middle line:90% a lantern in my face? 00:57:12.640 --> 00:57:14.305 align:middle line:90% It's like, get to the point. 00:57:14.305 --> 00:57:16.180 align:middle line:84% Show me equations, show me what you're doing, 00:57:16.180 --> 00:57:18.070 align:middle line:84% show me the measurements you performed in your laboratory, 00:57:18.070 --> 00:57:20.680 align:middle line:84% show me the missing factor of 2 pi in Maxwell's equation. 00:57:20.680 --> 00:57:21.910 align:middle line:90% Do something. 00:57:21.910 --> 00:57:24.790 align:middle line:84% Calculate something, you lazy dog! 00:57:24.790 --> 00:57:26.470 align:middle line:84% I think what Minkowski was reacting 00:57:26.470 --> 00:57:29.140 align:middle line:84% to was this kind of-- not just a kind of everything's 00:57:29.140 --> 00:57:32.420 align:middle line:84% in a specific coordinate basis, it's not very unified, 00:57:32.420 --> 00:57:34.940 align:middle line:84% we would learn fancy techniques later. 00:57:34.940 --> 00:57:38.540 align:middle line:84% It's not just that he doesn't-- that Einstein doesn't scale out 00:57:38.540 --> 00:57:41.390 align:middle line:84% the speed of light to make things more dimensionless. 00:57:41.390 --> 00:57:44.420 align:middle line:84% These are all things that we would admire or come to take-- 00:57:44.420 --> 00:57:47.510 align:middle line:84% to consider a natural move to make because-- 00:57:47.510 --> 00:57:50.090 align:middle line:84% on the benefit of work by people like Minkowski. 00:57:50.090 --> 00:57:51.800 align:middle line:84% I think to Minkowski, this was just 00:57:51.800 --> 00:57:55.070 align:middle line:84% a lot of philosophical blah, blah, blah that wasn't 00:57:55.070 --> 00:57:57.350 align:middle line:90% advancing any new mathematics. 00:57:57.350 --> 00:58:01.790 align:middle line:84% It wasn't analyzing concrete experiments in any detail. 00:58:01.790 --> 00:58:03.830 align:middle line:84% It wasn't telling us anything new about things 00:58:03.830 --> 00:58:05.330 align:middle line:90% that other researchers had done. 00:58:05.330 --> 00:58:06.892 align:middle line:84% It was like, hey, I read some stuff 00:58:06.892 --> 00:58:09.350 align:middle line:84% and I think it's confusing, why don't people talk about it? 00:58:09.350 --> 00:58:11.210 align:middle line:84% I think that was the kind of response 00:58:11.210 --> 00:58:14.150 align:middle line:84% from many readers, which is why many people ignored it. 00:58:14.150 --> 00:58:16.282 align:middle line:84% Those who bothered reading past page 5 00:58:16.282 --> 00:58:17.990 align:middle line:84% said, oh, I've seen that equation before, 00:58:17.990 --> 00:58:19.115 align:middle line:90% that's just Lorenz's stuff. 00:58:19.115 --> 00:58:21.380 align:middle line:84% Oh, it's the Lorentz-Einstein theory. 00:58:21.380 --> 00:58:23.420 align:middle line:84% I think it's that kind of response. 00:58:23.420 --> 00:58:25.220 align:middle line:84% And for Minkowski, I think he would 00:58:25.220 --> 00:58:31.330 align:middle line:84% have seen an insufficient precision and clarity. 00:58:31.330 --> 00:58:33.550 align:middle line:84% And now that's a value judgment, and that 00:58:33.550 --> 00:58:36.040 align:middle line:84% depends on our own personality, on our own training, 00:58:36.040 --> 00:58:37.090 align:middle line:90% our own toolkit. 00:58:37.090 --> 00:58:38.860 align:middle line:84% So for Minkowski, if you're not talking 00:58:38.860 --> 00:58:41.020 align:middle line:84% about what happens when you perform rotations, 00:58:41.020 --> 00:58:42.520 align:middle line:84% there should be something invariant. 00:58:42.520 --> 00:58:45.580 align:middle line:84% Like that should be less than 0 in geometry. 00:58:45.580 --> 00:58:47.650 align:middle line:84% The distance between my house and the town square 00:58:47.650 --> 00:58:49.953 align:middle line:84% hasn't changed even if I rotate the street map. 00:58:49.953 --> 00:58:51.370 align:middle line:84% Like, "Who doesn't think of that?" 00:58:51.370 --> 00:58:54.460 align:middle line:84% I think would have been Minkowski's response. 00:58:54.460 --> 00:58:58.720 align:middle line:84% Once you think in a geometrical way, according to Minkowski, 00:58:58.720 --> 00:59:00.970 align:middle line:84% there should be unavoidable follow-up questions, 00:59:00.970 --> 00:59:03.160 align:middle line:84% and this lazy dog is too busy talking 00:59:03.160 --> 00:59:06.610 align:middle line:84% about, he doesn't like the way textbooks describe Maxwell, 00:59:06.610 --> 00:59:07.910 align:middle line:90% like I got no time for this. 00:59:07.910 --> 00:59:11.230 align:middle line:84% I think that would have been a pretty reasonable approximation 00:59:11.230 --> 00:59:12.550 align:middle line:90% to his first reaction. 00:59:12.550 --> 00:59:14.373 align:middle line:84% And then he's like, oh, actually, 00:59:14.373 --> 00:59:15.790 align:middle line:84% OK, the speed of light is constant 00:59:15.790 --> 00:59:18.110 align:middle line:84% and I can perform these very clever sets of rotations, 00:59:18.110 --> 00:59:20.110 align:middle line:84% and of course, therefore, there's an invariance, 00:59:20.110 --> 00:59:22.000 align:middle line:84% and of course, there's an absolute away. 00:59:22.000 --> 00:59:24.710 align:middle line:84% So he has his own-- he's on his own train. 00:59:24.710 --> 00:59:27.730 align:middle line:84% He's on a different reference frame, so to speak. 00:59:27.730 --> 00:59:30.220 align:middle line:84% And certain questions are the obvious next ones 00:59:30.220 --> 00:59:34.130 align:middle line:84% to ask because he has a certain set of tools and starting 00:59:34.130 --> 00:59:34.680 align:middle line:90% point. 00:59:34.680 --> 00:59:35.780 align:middle line:90% And that's true of-- 00:59:35.780 --> 00:59:38.750 align:middle line:84% I think of all of us, and we'll see a contrasting example 00:59:38.750 --> 00:59:40.298 align:middle line:84% in this next part of today's class, 00:59:40.298 --> 00:59:42.590 align:middle line:84% we'll see that over and over again throughout the term. 00:59:42.590 --> 00:59:44.780 align:middle line:84% Not only about relativity, but quantum theory, 00:59:44.780 --> 00:59:47.150 align:middle line:84% about cosmology, about particle physics. 00:59:47.150 --> 00:59:49.190 align:middle line:84% And here's a kind of early example 00:59:49.190 --> 00:59:50.660 align:middle line:84% where we can sink our teeth into it 00:59:50.660 --> 00:59:53.690 align:middle line:84% and say, these people weren't just misreading it, 00:59:53.690 --> 00:59:57.050 align:middle line:84% they were doing really cool productive stuff, some of which 00:59:57.050 --> 00:59:59.090 align:middle line:84% we still take advantage of today, 00:59:59.090 --> 01:00:01.100 align:middle line:84% and yet, what they thought they were doing 01:00:01.100 --> 01:00:03.790 align:middle line:84% is neither what we think we do, nor what the original authors 01:00:03.790 --> 01:00:04.790 align:middle line:90% thought they were doing. 01:00:04.790 --> 01:00:08.840 align:middle line:84% So we have this plasticity of meaning and interpretation 01:00:08.840 --> 01:00:11.842 align:middle line:84% even when our equations agree perfectly, let alone 01:00:11.842 --> 01:00:13.550 align:middle line:84% when we come up with different equations. 01:00:13.550 --> 01:00:15.230 align:middle line:90% I find that wonderful. 01:00:15.230 --> 01:00:15.810 align:middle line:90% I love that. 01:00:15.810 --> 01:00:17.930 align:middle line:84% And in fact, they're all writing down the same factor gamma, 01:00:17.930 --> 01:00:19.763 align:middle line:84% that's fantastic, because then we can really 01:00:19.763 --> 01:00:22.760 align:middle line:84% map what is the same and what's not the same conceptually, 01:00:22.760 --> 01:00:25.197 align:middle line:84% let alone in the exchange of light signals. 01:00:25.197 --> 01:00:26.780 align:middle line:84% So keep that kind of question in mind. 01:00:26.780 --> 01:00:28.363 align:middle line:84% We'll come back to that theme, really, 01:00:28.363 --> 01:00:30.510 align:middle line:84% over and over this semester, it's a great question. 01:00:30.510 --> 01:00:32.980 align:middle line:84% Let me go on to the last part and look at our favorite 01:00:32.980 --> 01:00:35.230 align:middle line:84% Cambridge Wranglers, or at least my favorite Cambridge 01:00:35.230 --> 01:00:36.850 align:middle line:90% Wranglers. 01:00:36.850 --> 01:00:39.460 align:middle line:84% So, Minkowski wasn't the only person 01:00:39.460 --> 01:00:42.250 align:middle line:84% who slowly, sometimes kind of grudgingly, 01:00:42.250 --> 01:00:45.490 align:middle line:84% began to read Einstein's work on the electrodynamics of moving 01:00:45.490 --> 01:00:46.400 align:middle line:90% bodies. 01:00:46.400 --> 01:00:47.890 align:middle line:84% Another group that really did dig 01:00:47.890 --> 01:00:50.315 align:middle line:84% in in pretty substantive ways was a subset 01:00:50.315 --> 01:00:51.940 align:middle line:84% of these Cambridge Wranglers, the folks 01:00:51.940 --> 01:00:53.710 align:middle line:84% that now we've talked about a few times 01:00:53.710 --> 01:00:55.120 align:middle line:90% at Cambridge University. 01:00:55.120 --> 01:00:57.430 align:middle line:84% For them, much like for Minkowski, there 01:00:57.430 --> 01:01:00.490 align:middle line:84% were elements of real value in Einstein's work, 01:01:00.490 --> 01:01:02.740 align:middle line:84% they just weren't the elements that Minkowski 01:01:02.740 --> 01:01:05.270 align:middle line:84% considered valuable or that Einstein considered valuable. 01:01:05.270 --> 01:01:07.353 align:middle line:84% So let's take a little look at the kinds of things 01:01:07.353 --> 01:01:09.400 align:middle line:84% that they did with Einstein's paper. 01:01:09.400 --> 01:01:10.990 align:middle line:84% As a warm-up to make sense of this-- 01:01:10.990 --> 01:01:12.698 align:middle line:84% and by the way, for this part, of course, 01:01:12.698 --> 01:01:15.010 align:middle line:84% I'm drawing very heavily on the article in the reading 01:01:15.010 --> 01:01:17.290 align:middle line:90% for today from Andy Warwick. 01:01:17.290 --> 01:01:19.240 align:middle line:84% And I'll talk through some of the specifics-- 01:01:19.240 --> 01:01:21.610 align:middle line:84% some of the details of that article 01:01:21.610 --> 01:01:23.068 align:middle line:84% might be pretty confusing, so we'll 01:01:23.068 --> 01:01:25.360 align:middle line:84% talk through what were they doing at least a little bit 01:01:25.360 --> 01:01:25.870 align:middle line:90% more. 01:01:25.870 --> 01:01:28.240 align:middle line:84% The historical point I want to clarify 01:01:28.240 --> 01:01:30.760 align:middle line:84% is really just like Minkowski in the sense 01:01:30.760 --> 01:01:34.630 align:middle line:84% that very smart professional researchers were making sense 01:01:34.630 --> 01:01:37.160 align:middle line:84% of Einstein's work differently than Einstein did, 01:01:37.160 --> 01:01:39.200 align:middle line:84% and that didn't make them wrong, it 01:01:39.200 --> 01:01:42.110 align:middle line:84% made them-- it indicates that they were finding 01:01:42.110 --> 01:01:44.150 align:middle line:84% elements of value and of interest in Einstein's 01:01:44.150 --> 01:01:46.430 align:middle line:84% work different from Einstein's own and sometimes different 01:01:46.430 --> 01:01:46.972 align:middle line:90% from our own. 01:01:46.972 --> 01:01:48.688 align:middle line:90% That's the historical lesson. 01:01:48.688 --> 01:01:50.480 align:middle line:84% And if you get a little bit lost with image 01:01:50.480 --> 01:01:53.450 align:middle line:84% charges and conformal transformations, that's OK. 01:01:53.450 --> 01:01:55.160 align:middle line:84% In this class especially, that's OK. 01:01:55.160 --> 01:01:57.080 align:middle line:84% But I think it's still fun to see what were they really 01:01:57.080 --> 01:01:58.788 align:middle line:84% doing, so that's what we're going to talk 01:01:58.788 --> 01:02:00.205 align:middle line:90% about a little bit here. 01:02:00.205 --> 01:02:01.580 align:middle line:84% So many of you, again, might have 01:02:01.580 --> 01:02:05.000 align:middle line:84% seen, even maybe in high school classes on electricity 01:02:05.000 --> 01:02:08.120 align:middle line:84% and magnetism, there are all these really clever tricks 01:02:08.120 --> 01:02:10.880 align:middle line:84% that we still take advantage of to try 01:02:10.880 --> 01:02:13.880 align:middle line:84% to make pretty complicated problems simpler and yet 01:02:13.880 --> 01:02:15.710 align:middle line:90% can still get exact solutions. 01:02:15.710 --> 01:02:18.170 align:middle line:84% One of those that we learned pretty early on 01:02:18.170 --> 01:02:22.310 align:middle line:84% is called the method of image charges in electrostatics. 01:02:22.310 --> 01:02:25.580 align:middle line:84% So let's say some meanie, one of my mean old colleagues 01:02:25.580 --> 01:02:28.040 align:middle line:84% in the Department of Physics, assigned to you the problem 01:02:28.040 --> 01:02:31.370 align:middle line:84% to solve for the state of the electric field 01:02:31.370 --> 01:02:36.530 align:middle line:84% between a stationary point charge with a positive charge, 01:02:36.530 --> 01:02:39.860 align:middle line:84% some charge plus q, and some infinite 01:02:39.860 --> 01:02:43.340 align:middle line:84% in extent, some infinitely long, perfectly flat conducting 01:02:43.340 --> 01:02:45.260 align:middle line:90% plate that grounded. 01:02:45.260 --> 01:02:48.830 align:middle line:84% That's a pain in the neck to find the exact value 01:02:48.830 --> 01:02:49.910 align:middle line:90% of the electric field. 01:02:49.910 --> 01:02:53.000 align:middle line:84% It's a vector quantity to worry about how 01:02:53.000 --> 01:02:54.570 align:middle line:90% it's changing through space. 01:02:54.570 --> 01:02:56.690 align:middle line:84% So one thing that you may remember 01:02:56.690 --> 01:02:58.670 align:middle line:84% from studying Maxwell's equations 01:02:58.670 --> 01:03:00.680 align:middle line:84% is that for this kind of scenario, the field 01:03:00.680 --> 01:03:04.220 align:middle line:84% lines of the electric field all must intersect the plate 01:03:04.220 --> 01:03:05.300 align:middle line:90% at right angles. 01:03:05.300 --> 01:03:07.260 align:middle line:84% That's to remain consistent with the fact 01:03:07.260 --> 01:03:09.260 align:middle line:84% that this is a grounded conducting plate. 01:03:09.260 --> 01:03:11.300 align:middle line:84% If they didn't all intersect at right angles, 01:03:11.300 --> 01:03:13.430 align:middle line:84% you would actually build up a voltage difference, 01:03:13.430 --> 01:03:15.650 align:middle line:84% it would not satisfy the boundary conditions 01:03:15.650 --> 01:03:16.550 align:middle line:90% of the problem. 01:03:16.550 --> 01:03:18.020 align:middle line:90% It's a pain in the neck. 01:03:18.020 --> 01:03:18.830 align:middle line:90% Meanies. 01:03:18.830 --> 01:03:20.720 align:middle line:84% So there's a pretty clever trick. 01:03:20.720 --> 01:03:24.290 align:middle line:84% We can solve a much, much, much simpler mathematical setup 01:03:24.290 --> 01:03:27.350 align:middle line:84% to get the exact solution for the original problem. 01:03:27.350 --> 01:03:28.620 align:middle line:90% Forget about the plate. 01:03:28.620 --> 01:03:31.640 align:middle line:84% Throw the plate away for a few steps. 01:03:31.640 --> 01:03:34.520 align:middle line:84% And instead, insert a second point charge 01:03:34.520 --> 01:03:36.500 align:middle line:90% of equal but opposite sign. 01:03:36.500 --> 01:03:38.510 align:middle line:84% Equal-equal magnitude, but opposite sign. 01:03:38.510 --> 01:03:40.850 align:middle line:84% So now in this new scenario, we pretend 01:03:40.850 --> 01:03:45.193 align:middle line:84% there's nothing physical here, but we have an equal distance 01:03:45.193 --> 01:03:46.610 align:middle line:84% away from that where the plate had 01:03:46.610 --> 01:03:47.990 align:middle line:90% been in the original problem. 01:03:47.990 --> 01:03:51.380 align:middle line:84% We put down an imaginary image charge 01:03:51.380 --> 01:03:55.100 align:middle line:84% that has a negative charge, but at the same magnitude 01:03:55.100 --> 01:03:57.750 align:middle line:90% as the original positive charge. 01:03:57.750 --> 01:03:59.990 align:middle line:90% Now we can easily solve for-- 01:03:59.990 --> 01:04:01.640 align:middle line:84% much more easily solve for the value 01:04:01.640 --> 01:04:02.960 align:middle line:90% of the electrostatic potential. 01:04:02.960 --> 01:04:04.580 align:middle line:90% It's just two bodies. 01:04:04.580 --> 01:04:07.550 align:middle line:84% It just goes like a 1 over r between them. 01:04:07.550 --> 01:04:10.310 align:middle line:84% We can take the gradient, see how that changes through space. 01:04:10.310 --> 01:04:11.840 align:middle line:84% We can do everything we would ask 01:04:11.840 --> 01:04:15.680 align:middle line:84% to do to solve for the exact behavior of the electric field 01:04:15.680 --> 01:04:19.430 align:middle line:84% by replacing one problem with a simpler but equivalent one. 01:04:19.430 --> 01:04:21.230 align:middle line:84% That's the method of image charges. 01:04:21.230 --> 01:04:25.020 align:middle line:84% You can learn about that on Wikipedia or on any textbook. 01:04:25.020 --> 01:04:27.650 align:middle line:84% It's a standard technique that goes back really centuries. 01:04:27.650 --> 01:04:29.190 align:middle line:90% It's pretty powerful. 01:04:29.190 --> 01:04:30.855 align:middle line:90% Well, that's baby stuff. 01:04:30.855 --> 01:04:32.480 align:middle line:84% We're talking about Cambridge Wranglers 01:04:32.480 --> 01:04:34.340 align:middle line:84% here, or for that matter, MIT students. 01:04:34.340 --> 01:04:35.930 align:middle line:84% We don't have to do one single point 01:04:35.930 --> 01:04:39.050 align:middle line:84% charge behind what would have been an infinite unbending 01:04:39.050 --> 01:04:39.800 align:middle line:90% conducting plate. 01:04:39.800 --> 01:04:41.130 align:middle line:90% That is boring. 01:04:41.130 --> 01:04:42.628 align:middle line:90% Let's do some crazy stuff. 01:04:42.628 --> 01:04:44.420 align:middle line:84% And that's what the Wranglers got very good 01:04:44.420 --> 01:04:47.180 align:middle line:84% at doing with their personal tutors for the Tripos exam. 01:04:47.180 --> 01:04:49.250 align:middle line:84% So a very similar related technique 01:04:49.250 --> 01:04:51.530 align:middle line:84% became known as the method of inversion. 01:04:51.530 --> 01:04:54.500 align:middle line:84% And that's, again, really just mapping a difficult problem 01:04:54.500 --> 01:04:55.460 align:middle line:90% into a simpler one. 01:04:55.460 --> 01:04:57.800 align:middle line:84% And again, this illustration is taken 01:04:57.800 --> 01:05:00.200 align:middle line:84% from Andy Warwick's article, which 01:05:00.200 --> 01:05:02.330 align:middle line:84% then is elaborated upon in this book of his 01:05:02.330 --> 01:05:05.030 align:middle line:84% that I really can't recommend strongly enough. 01:05:05.030 --> 01:05:06.740 align:middle line:84% I love this book, Masters of Theory. 01:05:06.740 --> 01:05:10.670 align:middle line:84% But the article itself really gets this part across. 01:05:10.670 --> 01:05:12.808 align:middle line:84% Let's first do a little geometry, a little mapping 01:05:12.808 --> 01:05:14.600 align:middle line:84% exercise, which is what the Wranglers would 01:05:14.600 --> 01:05:17.570 align:middle line:84% have learned to do very early with their coaches. 01:05:17.570 --> 01:05:20.990 align:middle line:84% Take any single point A. Here's the point a for our case. 01:05:20.990 --> 01:05:25.130 align:middle line:84% Any point that happens to be inside a circle of radius k. 01:05:25.130 --> 01:05:28.400 align:middle line:84% So the point A is contained within that circle. 01:05:28.400 --> 01:05:32.207 align:middle line:84% We can then do a unique mapping to an inverse point, 01:05:32.207 --> 01:05:34.040 align:middle line:84% kind of like that-- where we place our image 01:05:34.040 --> 01:05:35.970 align:middle line:90% charge from the past example. 01:05:35.970 --> 01:05:39.560 align:middle line:84% There is a one-to-one map for a point corresponding uniquely 01:05:39.560 --> 01:05:42.560 align:middle line:84% to that point A that's now outside the circle. 01:05:42.560 --> 01:05:45.770 align:middle line:84% And so the way we determine where that point will be 01:05:45.770 --> 01:05:48.590 align:middle line:84% is by requiring the distance from the center of the circle 01:05:48.590 --> 01:05:50.120 align:middle line:84% to our original point, that distance 01:05:50.120 --> 01:05:54.290 align:middle line:84% OA, times the distance between the center of the circle 01:05:54.290 --> 01:05:56.270 align:middle line:84% and the new point, the inverse point. 01:05:56.270 --> 01:05:58.220 align:middle line:84% The product of those distances must equal 01:05:58.220 --> 01:05:59.690 align:middle line:90% the square of this radius. 01:05:59.690 --> 01:06:01.130 align:middle line:84% That's enough to do this mapping. 01:06:01.130 --> 01:06:03.770 align:middle line:84% It's actually called a conformal mapping. 01:06:03.770 --> 01:06:06.890 align:middle line:84% Now what if there were some point that point A is actually 01:06:06.890 --> 01:06:10.940 align:middle line:84% moving along this curve, this orange curve between points P 01:06:10.940 --> 01:06:12.020 align:middle line:90% and Q? 01:06:12.020 --> 01:06:15.200 align:middle line:84% We can imagine, there is a whole set of points, each of which 01:06:15.200 --> 01:06:17.750 align:middle line:84% lies entirely within this original circle. 01:06:17.750 --> 01:06:20.750 align:middle line:84% For each of these points along the orange curve between p 01:06:20.750 --> 01:06:22.460 align:middle line:90% and q, we can do the same trick. 01:06:22.460 --> 01:06:26.180 align:middle line:84% We can map them to each of their corresponding inverse points. 01:06:26.180 --> 01:06:29.050 align:middle line:84% For every single point along this orange curve, 01:06:29.050 --> 01:06:31.950 align:middle line:84% we require this relationship to hold. 01:06:31.950 --> 01:06:34.010 align:middle line:84% And we map them to an inverse arc, 01:06:34.010 --> 01:06:37.070 align:middle line:84% a collection of inverse points that are entirely 01:06:37.070 --> 01:06:38.460 align:middle line:90% outside the circle. 01:06:38.460 --> 01:06:41.420 align:middle line:84% So if we imagine swooping from point p 01:06:41.420 --> 01:06:44.360 align:middle line:84% to q within that circle, the mapped motion, 01:06:44.360 --> 01:06:46.310 align:middle line:84% the uniquely corresponding motion 01:06:46.310 --> 01:06:50.000 align:middle line:84% would swoop from point p prime to q prime. 01:06:50.000 --> 01:06:51.560 align:middle line:84% Now this transformation is called 01:06:51.560 --> 01:06:53.117 align:middle line:90% a conformal transformation. 01:06:53.117 --> 01:06:54.200 align:middle line:90% We use this all the time-- 01:06:54.200 --> 01:06:56.840 align:middle line:84% I use these all the time, even to this day in my own physics 01:06:56.840 --> 01:06:57.380 align:middle line:90% research. 01:06:57.380 --> 01:06:59.630 align:middle line:84% They're really, really helpful in topics 01:06:59.630 --> 01:07:01.610 align:middle line:84% like general relativity and cosmology. 01:07:01.610 --> 01:07:04.620 align:middle line:84% The Wranglers are doing this for electrostatics 01:07:04.620 --> 01:07:06.540 align:middle line:90% to get ready for their exam. 01:07:06.540 --> 01:07:10.080 align:middle line:84% So this transformation preserves angles, the angles between any 01:07:10.080 --> 01:07:12.060 align:middle line:84% of these arcs, any of these line segments, 01:07:12.060 --> 01:07:13.800 align:middle line:90% this angle marked w here. 01:07:13.800 --> 01:07:16.380 align:middle line:84% Those angles are preserved, the lengths are not. 01:07:16.380 --> 01:07:19.530 align:middle line:84% You can see, the line segment pq is clearly 01:07:19.530 --> 01:07:22.530 align:middle line:84% a shorter overall length-- the distance between points p and q 01:07:22.530 --> 01:07:25.950 align:middle line:84% is shorter than the distance between the corresponding 01:07:25.950 --> 01:07:28.390 align:middle line:84% inverse points p prime and q prime. 01:07:28.390 --> 01:07:30.840 align:middle line:84% So very generally, conformal transformations 01:07:30.840 --> 01:07:32.880 align:middle line:84% preserve angles, but not lengths. 01:07:32.880 --> 01:07:35.377 align:middle line:84% If that sounds confusing to you, you're in good company. 01:07:35.377 --> 01:07:36.960 align:middle line:84% The point is, that's the kind of stuff 01:07:36.960 --> 01:07:39.168 align:middle line:84% these Wranglers would have learned with their coaches 01:07:39.168 --> 01:07:39.930 align:middle line:90% pretty early. 01:07:39.930 --> 01:07:42.630 align:middle line:84% There are properties of these geometrical transformations, 01:07:42.630 --> 01:07:48.070 align:middle line:84% the upshot of which is to make complicated problems easier. 01:07:48.070 --> 01:07:50.790 align:middle line:84% So now let's get to an actual Tripos-type problem. 01:07:50.790 --> 01:07:55.020 align:middle line:84% Now let's say, because you're at Cambridge in 1880 or 1890, 01:07:55.020 --> 01:07:56.880 align:middle line:84% they don't ask you to solve for the field 01:07:56.880 --> 01:07:59.920 align:middle line:84% between a static positive charge and an infinite conducting 01:07:59.920 --> 01:08:00.420 align:middle line:90% plate. 01:08:00.420 --> 01:08:01.560 align:middle line:90% Come on. 01:08:01.560 --> 01:08:02.460 align:middle line:90% We are Wranglers. 01:08:02.460 --> 01:08:04.860 align:middle line:84% Instead, we want to solve for the electric field lines, 01:08:04.860 --> 01:08:07.320 align:middle line:84% or let's say the equipotential surfaces where 01:08:07.320 --> 01:08:09.870 align:middle line:84% the electrostatic potential is equal-- 01:08:09.870 --> 01:08:11.520 align:middle line:90% so equipotential surface. 01:08:11.520 --> 01:08:14.880 align:middle line:84% Due to a single charge at location 01:08:14.880 --> 01:08:18.960 align:middle line:84% O that's near a grounded conducting sphere centered 01:08:18.960 --> 01:08:22.590 align:middle line:84% at B. So here is the actual physical parts of our problem. 01:08:22.590 --> 01:08:25.830 align:middle line:84% Some charge here sitting at rest, electrostatics, 01:08:25.830 --> 01:08:28.529 align:middle line:84% some electric charge at point O that's 01:08:28.529 --> 01:08:31.439 align:middle line:84% near some spherical grounded conducting sphere-- 01:08:31.439 --> 01:08:34.290 align:middle line:84% there's a harder version than our infinite plane. 01:08:34.290 --> 01:08:36.450 align:middle line:84% Well, let's use this method of inversion 01:08:36.450 --> 01:08:38.800 align:middle line:84% to treat what turns out to be a much simpler problem. 01:08:38.800 --> 01:08:41.460 align:middle line:84% So we have to map the simpler version first. 01:08:41.460 --> 01:08:44.819 align:middle line:84% Instead of considering either point O or this conducting 01:08:44.819 --> 01:08:47.100 align:middle line:84% sphere B, we introduce the equivalent 01:08:47.100 --> 01:08:48.660 align:middle line:90% of a kind of image charge. 01:08:48.660 --> 01:08:52.800 align:middle line:84% We consider a charged conducting sphere centered at point A. 01:08:52.800 --> 01:08:54.090 align:middle line:90% We put in this-- 01:08:54.090 --> 01:08:58.109 align:middle line:84% like our image charge-- our new geometrical simplified system 01:08:58.109 --> 01:09:01.410 align:middle line:84% some positive charge and a sphere at A. Then 01:09:01.410 --> 01:09:04.600 align:middle line:84% we draw the field lines and the equipotential surfaces 01:09:04.600 --> 01:09:05.100 align:middle line:90% for that. 01:09:05.100 --> 01:09:06.310 align:middle line:90% Now that's very easy. 01:09:06.310 --> 01:09:07.319 align:middle line:90% It's just a sphere. 01:09:07.319 --> 01:09:09.960 align:middle line:84% They're just going to be concentric circles around it 01:09:09.960 --> 01:09:11.430 align:middle line:90% for the equipotential surfaces. 01:09:11.430 --> 01:09:13.109 align:middle line:90% The field is just radial. 01:09:13.109 --> 01:09:15.960 align:middle line:84% It goes back to Faraday's lines of force. 01:09:15.960 --> 01:09:18.750 align:middle line:84% It couldn't be easier for these Wranglers. 01:09:18.750 --> 01:09:21.540 align:middle line:84% Now we use that method of inversion 01:09:21.540 --> 01:09:23.399 align:middle line:84% to map it back to the original problem. 01:09:23.399 --> 01:09:25.649 align:middle line:84% We're going to close this imaginary object 01:09:25.649 --> 01:09:30.430 align:middle line:84% A in a sphere of radius K. And they 01:09:30.430 --> 01:09:32.979 align:middle line:84% will map every point along this orange curve 01:09:32.979 --> 01:09:37.189 align:middle line:84% to the corresponding image points outside of our circle. 01:09:37.189 --> 01:09:39.760 align:middle line:84% And now we've mapped the equipotential surfaces 01:09:39.760 --> 01:09:42.439 align:middle line:84% for the original problem based on this much simplified 01:09:42.439 --> 01:09:42.939 align:middle line:90% problem. 01:09:42.939 --> 01:09:46.330 align:middle line:84% If that's confusing to you, first of all, good. 01:09:46.330 --> 01:09:48.550 align:middle line:90% You've led a healthy life. 01:09:48.550 --> 01:09:49.970 align:middle line:90% This is not on the exam. 01:09:49.970 --> 01:09:51.649 align:middle line:84% This is just meant to be an example 01:09:51.649 --> 01:09:53.290 align:middle line:84% to make sure that the kind of reasoning 01:09:53.290 --> 01:09:54.790 align:middle line:84% that these Wranglers would have been 01:09:54.790 --> 01:09:56.980 align:middle line:84% immersed in of the sort we've seen a few times. 01:09:56.980 --> 01:09:59.750 align:middle line:84% This is like standard Tripos stuff. 01:09:59.750 --> 01:10:02.650 align:middle line:84% And remember, unlike us here at MIT, 01:10:02.650 --> 01:10:05.260 align:middle line:84% the undergraduates at Cambridge in the 1880s 01:10:05.260 --> 01:10:08.158 align:middle line:84% would have done all this on a timed exam that 01:10:08.158 --> 01:10:10.450 align:middle line:84% determined their graduation rank and would be published 01:10:10.450 --> 01:10:12.490 align:middle line:84% in the national newspaper, so we won't to ask that of you. 01:10:12.490 --> 01:10:14.320 align:middle line:84% So now let's come back to what Andy is 01:10:14.320 --> 01:10:16.060 align:middle line:90% writing about in this article. 01:10:16.060 --> 01:10:19.120 align:middle line:84% This method of inversion and these geometrical techniques, 01:10:19.120 --> 01:10:20.680 align:middle line:84% like conformal transformations, that 01:10:20.680 --> 01:10:25.630 align:middle line:84% was like daily stuff for these Cambridge Wranglers by 1900. 01:10:25.630 --> 01:10:27.970 align:middle line:84% These techniques apply to electrostatics. 01:10:27.970 --> 01:10:31.040 align:middle line:84% Imagine, you have a fixed charge sitting still 01:10:31.040 --> 01:10:34.850 align:middle line:84% at point O, a fixed grounded sphere, nothing 01:10:34.850 --> 01:10:36.500 align:middle line:90% changes over time. 01:10:36.500 --> 01:10:38.840 align:middle line:84% And that is to say a little more quantitatively, 01:10:38.840 --> 01:10:40.520 align:middle line:84% you're always solving for situations 01:10:40.520 --> 01:10:43.130 align:middle line:84% in which the electric potential is not varying. 01:10:43.130 --> 01:10:45.740 align:middle line:84% It's a problem in electrostatics not varying over time 01:10:45.740 --> 01:10:47.940 align:middle line:84% even though it varies over space. 01:10:47.940 --> 01:10:51.200 align:middle line:84% So what these young recent Wranglers wanted to do, 01:10:51.200 --> 01:10:53.330 align:middle line:84% Ebenezer Cunningham and Harry Bateman, 01:10:53.330 --> 01:10:55.460 align:middle line:84% was to actually use Einstein's work 01:10:55.460 --> 01:10:58.260 align:middle line:84% to tackle a problem they cared about, 01:10:58.260 --> 01:11:01.400 align:middle line:84% which is to generalize these inversion problems 01:11:01.400 --> 01:11:04.520 align:middle line:84% and conformal transformations to situations where objects might 01:11:04.520 --> 01:11:07.340 align:middle line:84% be moving around to electrodynamics of moving 01:11:07.340 --> 01:11:09.200 align:middle line:90% bodies, not to electrostatics. 01:11:09.200 --> 01:11:12.283 align:middle line:84% You can see why Einstein-- the title of Einstein's paper 01:11:12.283 --> 01:11:13.700 align:middle line:84% would have caught their attention. 01:11:13.700 --> 01:11:15.860 align:middle line:84% They wanted to understand how to generalize 01:11:15.860 --> 01:11:18.140 align:middle line:84% these Wrangler-like techniques to the question 01:11:18.140 --> 01:11:20.365 align:middle line:84% of the electrodynamics of moving bodies. 01:11:20.365 --> 01:11:21.740 align:middle line:84% And they thought Einstein's paper 01:11:21.740 --> 01:11:24.200 align:middle line:84% had some pretty clever tools in there to do that. 01:11:24.200 --> 01:11:27.470 align:middle line:84% So they could identify, using Einstein's work, all 01:11:27.470 --> 01:11:30.080 align:middle line:84% the transformations, which we would now call the Lorentz 01:11:30.080 --> 01:11:31.430 align:middle line:90% transformations, lambda-- 01:11:31.430 --> 01:11:32.983 align:middle line:84% Einstein had separately rederived 01:11:32.983 --> 01:11:34.400 align:middle line:84% those in a later part of the paper 01:11:34.400 --> 01:11:35.870 align:middle line:84% that I didn't assign for our class, 01:11:35.870 --> 01:11:37.993 align:middle line:84% but it's in Einstein's paper, too. 01:11:37.993 --> 01:11:39.410 align:middle line:84% These are the transformations that 01:11:39.410 --> 01:11:42.450 align:middle line:84% would leave the original wave equation invariant. 01:11:42.450 --> 01:11:45.320 align:middle line:84% So then they could use these inversion, these mapping 01:11:45.320 --> 01:11:48.140 align:middle line:84% techniques, the characteristic Wrangler techniques 01:11:48.140 --> 01:11:50.630 align:middle line:84% to map into new dynamical solutions 01:11:50.630 --> 01:11:53.090 align:middle line:84% that could vary in time and not just space, 01:11:53.090 --> 01:11:55.640 align:middle line:84% keeping in mind that now the time coordinate, as well 01:11:55.640 --> 01:11:59.040 align:middle line:84% as the spatial coordinates, might need to be shifted thanks 01:11:59.040 --> 01:12:00.700 align:middle line:90% to this Lorentz transformation. 01:12:00.700 --> 01:12:02.970 align:middle line:84% So they wanted to use Einstein's paper 01:12:02.970 --> 01:12:06.930 align:middle line:84% to find the most general class of transformations lambda that 01:12:06.930 --> 01:12:11.220 align:middle line:84% would leave this harder problem invariant. 01:12:11.220 --> 01:12:13.770 align:middle line:84% To do for electrodynamics what they had already 01:12:13.770 --> 01:12:17.082 align:middle line:84% learned with their coaches to do for electrostatics. 01:12:17.082 --> 01:12:19.290 align:middle line:84% That's what relativity was all about for these folks. 01:12:19.290 --> 01:12:21.810 align:middle line:84% They were reading Einstein's work very carefully. 01:12:21.810 --> 01:12:23.970 align:middle line:84% They were doing extremely productive, hard, 01:12:23.970 --> 01:12:24.842 align:middle line:90% original work. 01:12:24.842 --> 01:12:26.550 align:middle line:84% They just were doing things that Einstein 01:12:26.550 --> 01:12:29.250 align:middle line:84% didn't notice or care or think was important 01:12:29.250 --> 01:12:30.930 align:middle line:90% or that Minkowski did. 01:12:30.930 --> 01:12:33.600 align:middle line:84% So they weren't trying to understand the ether, per se. 01:12:33.600 --> 01:12:36.180 align:middle line:84% Again, they weren't trying to understand Machian positivism. 01:12:36.180 --> 01:12:38.070 align:middle line:84% They weren't even really that concerned 01:12:38.070 --> 01:12:39.720 align:middle line:90% about Minkowski space-time. 01:12:39.720 --> 01:12:41.550 align:middle line:84% For them, it's about wrangling most 01:12:41.550 --> 01:12:45.660 align:middle line:84% of the manipulations of mapping complicated problems 01:12:45.660 --> 01:12:47.940 align:middle line:84% into simpler ones, and Einstein's paper then 01:12:47.940 --> 01:12:51.730 align:middle line:84% gave them additional tools with which to do their problem. 01:12:51.730 --> 01:12:54.060 align:middle line:84% So they weren't ignoring Einstein's work, 01:12:54.060 --> 01:12:56.628 align:middle line:84% but they also weren't card-carrying Einsteineans. 01:12:56.628 --> 01:12:58.170 align:middle line:84% So that's what I wanted to go through 01:12:58.170 --> 01:13:01.350 align:middle line:84% just to make sure that the real historical point of Andy's 01:13:01.350 --> 01:13:02.642 align:middle line:90% paper would be clear. 01:13:02.642 --> 01:13:05.100 align:middle line:84% It gets pretty complicated, the details of these inversions 01:13:05.100 --> 01:13:06.090 align:middle line:90% and all that. 01:13:06.090 --> 01:13:07.598 align:middle line:84% I'd be glad to chat more about that. 01:13:07.598 --> 01:13:09.390 align:middle line:84% But the main point-- the main lesson for us 01:13:09.390 --> 01:13:12.060 align:middle line:84% is really that they're also doing real work 01:13:12.060 --> 01:13:15.270 align:middle line:84% with Einstein's paper, just different work than Einstein. 01:13:15.270 --> 01:13:17.610 align:middle line:90% So let me wrap up for today. 01:13:17.610 --> 01:13:20.250 align:middle line:84% Researchers didn't just read Einstein's paper 01:13:20.250 --> 01:13:21.930 align:middle line:84% and become convinced, they didn't 01:13:21.930 --> 01:13:24.360 align:middle line:84% become like converted Einsteineans 01:13:24.360 --> 01:13:25.800 align:middle line:90% for a long, long time. 01:13:25.800 --> 01:13:28.230 align:middle line:84% Most people ignored the paper altogether. 01:13:28.230 --> 01:13:30.480 align:middle line:84% Those few who paid attention often 01:13:30.480 --> 01:13:34.140 align:middle line:84% thought it was just a minor elaboration of previous work. 01:13:34.140 --> 01:13:37.020 align:middle line:84% The few who really did pay attention to it much more 01:13:37.020 --> 01:13:39.840 align:middle line:84% squarely did so from within their own context. 01:13:39.840 --> 01:13:43.110 align:middle line:84% We can even jokingly say, within their own frames of reference, 01:13:43.110 --> 01:13:45.660 align:middle line:84% much like Einstein's favorite example 01:13:45.660 --> 01:13:48.240 align:middle line:84% of observers on the platform or on the train. 01:13:48.240 --> 01:13:49.950 align:middle line:90% They do different stuff with it. 01:13:49.950 --> 01:13:52.860 align:middle line:84% Different parts of the paper are relevant to them and of value. 01:13:52.860 --> 01:13:54.870 align:middle line:84% They're not misunderstanding Einstein's work, 01:13:54.870 --> 01:13:56.760 align:middle line:84% they're like utterly understanding it. 01:13:56.760 --> 01:13:58.740 align:middle line:84% They're doing real stuff with it, just not what 01:13:58.740 --> 01:14:00.270 align:middle line:90% Einstein had first intended. 01:14:00.270 --> 01:14:03.060 align:middle line:84% So we saw Minkowski reinterpreted it 01:14:03.060 --> 01:14:06.360 align:middle line:84% in terms of a certain kind of geometrical vision. 01:14:06.360 --> 01:14:08.250 align:middle line:84% The Cambridge gang, Cunningham Bateman 01:14:08.250 --> 01:14:09.750 align:middle line:84% and some of their immediate circles, 01:14:09.750 --> 01:14:12.000 align:middle line:84% they do other kinds of geometrical things with it 01:14:12.000 --> 01:14:13.980 align:middle line:84% because they have other concerns more related 01:14:13.980 --> 01:14:15.840 align:middle line:90% to the properties and-- 01:14:15.840 --> 01:14:19.140 align:middle line:84% transformation properties of differential equations. 01:14:19.140 --> 01:14:20.910 align:middle line:84% The point is, none of these readers 01:14:20.910 --> 01:14:24.213 align:middle line:84% seem to care much about what was most important to Einstein. 01:14:24.213 --> 01:14:26.130 align:middle line:84% As far as Einstein was concerned, none of them 01:14:26.130 --> 01:14:27.960 align:middle line:84% were getting the point of his article 01:14:27.960 --> 01:14:29.752 align:middle line:84% even though these were among the people who 01:14:29.752 --> 01:14:31.290 align:middle line:90% paid any attention at all. 01:14:31.290 --> 01:14:34.050 align:middle line:84% Einstein had argued in those opening paragraphs 01:14:34.050 --> 01:14:36.150 align:middle line:84% that the ether was merely superfluous. 01:14:36.150 --> 01:14:37.680 align:middle line:90% Who cared? 01:14:37.680 --> 01:14:40.060 align:middle line:84% They didn't-- that wasn't the part that landed with them. 01:14:40.060 --> 01:14:43.260 align:middle line:84% Moreover, Einstein had insisted that we start with kinematics 01:14:43.260 --> 01:14:44.310 align:middle line:90% instead of dynamics. 01:14:44.310 --> 01:14:47.340 align:middle line:84% Forget-- that didn't wash, that didn't register. 01:14:47.340 --> 01:14:49.980 align:middle line:84% Instead, as we've seen before, and we'll see again, 01:14:49.980 --> 01:14:52.320 align:middle line:84% even the exact same equations could 01:14:52.320 --> 01:14:54.323 align:middle line:84% inspire quite different interpretations 01:14:54.323 --> 01:14:55.240 align:middle line:90% or different meanings. 01:14:55.240 --> 01:14:57.800 align:middle line:90% So I'll stop there. 01:14:57.800 --> 01:15:02.920 align:middle line:84% Any other questions on that last part about the Cambridge gang? 01:15:02.920 --> 01:15:05.590 align:middle line:84% And again, if the particulars of conformal transformations 01:15:05.590 --> 01:15:09.215 align:middle line:84% or inversion went by too quickly or Andy's paper was confusing, 01:15:09.215 --> 01:15:10.840 align:middle line:84% please don't worry, I'd be glad to chat 01:15:10.840 --> 01:15:12.715 align:middle line:84% more if it's of interest for you, 01:15:12.715 --> 01:15:14.440 align:middle line:84% but the historical point is just to say 01:15:14.440 --> 01:15:16.603 align:middle line:90% they were doing real work. 01:15:16.603 --> 01:15:17.770 align:middle line:90% It's still in our textbooks. 01:15:17.770 --> 01:15:19.090 align:middle line:90% It was valuable work. 01:15:19.090 --> 01:15:22.420 align:middle line:84% It was just different work than Einstein's or Minkowski's 01:15:22.420 --> 01:15:24.038 align:middle line:90% or Lorentz's and so on. 01:15:24.038 --> 01:15:26.080 align:middle line:84% That's really the point I think we can hold on to 01:15:26.080 --> 01:15:26.755 align:middle line:90% in this class. 01:15:26.755 --> 01:15:29.453 align:middle line:90% 01:15:29.453 --> 01:15:30.620 align:middle line:90% Any other questions on that? 01:15:30.620 --> 01:15:33.780 align:middle line:90% 01:15:33.780 --> 01:15:36.960 align:middle line:84% Anyone want to volunteer to do more inversion problems 01:15:36.960 --> 01:15:38.340 align:middle line:90% for conducting spheres and-- 01:15:38.340 --> 01:15:39.150 align:middle line:90% I hate that stuff. 01:15:39.150 --> 01:15:40.020 align:middle line:90% That stuff, yeah. 01:15:40.020 --> 01:15:40.832 align:middle line:90% Julia says no. 01:15:40.832 --> 01:15:41.790 align:middle line:90% I'm with Julia on this. 01:15:41.790 --> 01:15:45.917 align:middle line:84% That stuff makes me-- that really makes my head hurt. 01:15:45.917 --> 01:15:47.500 align:middle line:84% And you want to do that for five hours 01:15:47.500 --> 01:15:49.180 align:middle line:84% before lunch like a good Wrangler? 01:15:49.180 --> 01:15:50.830 align:middle line:90% That's another question for you. 01:15:50.830 --> 01:15:53.190 align:middle line:84% Your math homework before lunchtime. 01:15:53.190 --> 01:15:53.690 align:middle line:90% OK. 01:15:53.690 --> 01:15:56.980 align:middle line:84% If there's no more questions for now, then I'll pause there. 01:15:56.980 --> 01:15:58.447 align:middle line:84% We'll pick it up early next week. 01:15:58.447 --> 01:16:01.030 align:middle line:84% And we'll-- and we'll get into many of the questions that were 01:16:01.030 --> 01:16:03.070 align:middle line:90% raised even for today. 01:16:03.070 --> 01:16:05.380 align:middle line:84% What is Einstein himself do with all this work 01:16:05.380 --> 01:16:07.810 align:middle line:84% in his march toward what we would eventually 01:16:07.810 --> 01:16:09.848 align:middle line:84% call the General Theory of Relativity? 01:16:09.848 --> 01:16:10.765 align:middle line:90% So we'll come to that. 01:16:10.765 --> 01:16:14.520 align:middle line:84% That will occupy us, then, for our next class. 01:16:14.520 --> 01:16:21.000 align:middle line:90%